)^^JJLJl   ^.  ''li^^v.^xJLA^ 


AN    INTRODUCTION 

TO 

PROJECTIVE  GEOMETRY 


AND    ITS 


APPLICATIONS 


AN    ANALYTIC    AND    SYNTHETIC 
TREATMENT 


\ 

ARNOLD    EMCH,   Ph.D., 

Professor  of  Graphics  and  Mathematics  in  the  Ujiiversity  of  Colorado 


F/JCSr    EDITION 
FIRST     THOUSAND 


NEW  YORK 

JOHN   WILEY   &    SONS 

London:   CHAPMAN   &    HALL,  Limited 

1905 


BOSTON    COLLEGE    LlBRARf 
CHESTNUT  HILL,  MASS. 


)y\jJUJl   ^.  '"li^'v.^xJlla^ 


AN    INTRODUCTION 

TO 

PROJECTIVE  GEOMETRY 


AND    ITS 


APPLICATIONS 


AN    ANALYTIC    AND    SYNTHETIC 
TREATMENT 


ARNOLD    EMCH,   Ph.D., 


\ 


Professor  of  Graphics  and  Alatheviaiics  in  the  University  cf  Colorado 


FIRST    EDITION 
FIRST     THOUSAND 


NEW  YORK 

JOHN   WILEY   &    SONS 

London:   CHAPMAN   &    HALL,  Limited 

1905 


BOSTON    COLLEGE    LIBRARf 
CHESTNUT  HILL,  MASS. 


Copj'right,    1905, 

BY 

ARNOLD  EMCH. 


i 


ROBERT  DRUMMOND,    PRINTER,   NEW   YORK. 


PREFACE. 


Treatises  on  Projective  Geometry  are  usually  written  with 
the  object  of  presenting  this  science  in  a  purely  systematic  form; 
hardly  any  attention  is  paid  to  the  applications.  As  a  rule  the 
methods  of  "  arithmetized "  mathematics  have  to  be  transformed, 
made  more  concrete,  before  they  lend  themselves  to  the  solution 
of  practical  problems;  and  this,  in  the  judgment  of  many,  dis- 
figures the  purely  scientific  method. 

In  this  respect,  projective  geometry,  geometry  of  position,  is 
no  exception.  The  puristic  tendencies  of  von  Staudt,  Reye,  and 
others  culminate  in  the  modern  Italian  school  of  geometer-logicians, 
headed  by  Veronese  ^  and  Enriques.  The  latter's  projective 
geometry  ^  contains  an  admirable  logical  presentation  of  the 
subject.  With  Enriques  projective  geometry  is  a  "visual"  sci- 
ence, and  everything  is  foreign  to  it  which  cannot  be  based  upon 
the  axioms  of  vision. 

It  seems  doubtful  whether  the  axioms  of  vision  alone  can 
establish  a  sound  projective  geometry.  Enriques  himself,  in 
his  book,  lets  the  fundamental  elements  of  the  first  order  be 
generated  by  motion!  In  this  visual  geometry  metrical  proper- 
ties, which  are  indispensable  in  the  applications,  appear  as  special 
cases  and  are  of  secondary  importance.  Conies  result  from  the 
theory  of  polarity. 

On  the  other  hand,  Fiedler,  Wiener  and  others  show  that 
the  methods  of  Poncelet,  Steiner,  Chasles,  and  Cremona  naturally 

^  Grundzuge  der  Geometrie,  Teubner,  Leipzig. 
*  German  translation,  Teubner,  Leipzig. 


IV  PREFACE. 

follow  from  the  study  of  descriptive  geometry.  With  them, 
projective  and  descriptive  geometry  are  organically  related  and 
each  branch  benefits  by  its  connection  with  the  other.  Little 
attention  is  paid  to  the  so-called  foundations. 

As  the  present  book  has  been  written  with  a  utilitarian  pur- 
pose, considerable  space  is  given  to  the  applications;  and  in 
their  treatment  use  has  sometimes  been  made  of  original  analytic 
and  geometric  methods  of  attack  and  solution.  It  has  thus  been 
found  possible  to  include  some  new  subject-matter  and  especially 
certain  parts  of  modern  analytical  geometry. 

In  addition  to  the  traditional  contents  of  the  standard  ele- 
mentary treatises,  two  chapters  on  pencils  and  ranges  of  conies, 
including  cubics,  and  on  the  applications  in  mechanics  have 
been  added.  The  Steinerian  transformation  contained  in  Chap- 
ter IV,  in  connection  with  the  study  of  plane  cubics,  is  a  brilliant 
example  of  the  original  power  of  projective  geometry;  and  as  it 
is  elementary,  it  seems  natural  to  introduce  it  after  the  theory  of 
conies.  As  a  novel  feature  the  realization  of  collineations  by 
linkages,  described  in  Chapter  V,  may  be  mentioned. 

Much  time  may  profitably  be  devoted  to  the  original  prob- 
lems and  to  the  constructions  involved  in  them.  No  first  study 
of  projective  geometry  can  be  successful  without  the  constant 
use  of  ruler  and  compass. 

My  thanks  are  due  to  my  colleague,  Professor  Ira  M.  De  Long, 
for  many  valuable  suggestions  as  to  matters  of  form. 

Corrections  and  suggestions  as  to  either  the  form  or  the  matter 
of  the  text  are  earnestly  solicited. 

Arnold  Emch. 
Boulder,  Colorado, 
July,  1904. 


CONTENTS. 


CHAPTER  I. 


General   Considerations.    Anharmonic   Ratio.     Projective   Ranges 
AND  Pencils.     Polar  Involution  of  the  Circle. 

PAGE 

§    I.  Geometric  quantities  and  their  signs i 

§    2.  Anharmonic  ratio.     Projective  transformation  of  the  points  of  a 


straight  Hne. 


5 


§    3.  Involution u 

§    4.  Projective  pencils  of  rays i? 

§    5.  Involutoric  pencils 18 

§    6.  Product  of  projective  pencils  and  ranges 22 

§    7.  Exercises  and  problems 25 

§    8.  The  complete  quadrilateral 16 

§    9.  Perspective  pencils  and  ranges 29 

§  10.  General  construction  of  projective  pencils  and  ranges 30 

§11.  Exercises  and  problems 34 

§  12.  Projective  properties  of  the  circle ?5 

§  13.  Polar  involution  of  the  circle 38 

§  14.  Continuation  of  §  13 41 

CHAPTER  II. 

COLLINEATION. 

§  15.  Central  projection 45 

§  16.  Analytical  representation  of  central  projection 49 

§17.  Special  cases  of  central  projection 51 

§  18.  Exercises  and  problems 57 

§  19.  ColHneation 59 

§  20.  Geometrical  determination  and  discussion  of  collineation. 63 

V 


VI  CONTENTS. 

PAGE 

§  21.  Continuous  groups  of  projective  transformations 66 

§  22.  The  principle  of  duality 68 

§  23.  Exercises  and  problems 70 

§  24.  Orthographic  projection 72 

§  25.  Affinity  between  horizontal  and  vertical   projections   of  a   plane 

figure 77 

§  26.  Homologous  triangles. 80 

§  27.  A  few  applications  of  perspective 86 

§  28.  Exercises  and  problems 90 

CHAPTER  III. 

Theory  of  Conics. 

§  29.  Introduction » 92 

§  30.  Identity  of  curves  of  the  second  order  and  class  with  conics 93 

§  31.  Linear  transformation  of  a  curve  of  the  second  order 94 

§  32.  Polar  involution  of  conics.     Center.     Diameters.     Axes.     Asymp 

totes 102 

§  2^.  Existence  of  ellipse,  hyperbola,  parabola,  and  their  foci 104 

§  34.  Construction  of  foci  independent  of  central  projection 107 

§  35.  Focal  properties  of  conics 109 

§  36.  Analytical  expression  for  tangents  and  polars.     Equation  in  line- 
coordinates 118 

§  37.  Theory  of  reciprocal  polars 123 

§  38.  General  reciprocal  transformation.     Polar  systems 126 

§  39.  Theorems  of  Pascal  and  Brianchon 133 

§  40.  Applications  of  Pascal's  and  Brianchon's  theorems 137 

§  41.  Conics  in  mechanical  drawing  and  perspective 140 

§42.  Special  constructions  of  conics  by  central  projection  and  parallel 

projection 146 

§  43.  Problems  of  the  second  order 161 

§  44.  An  optical  problem 167 

CHAPTER  IV. 

Pencils  and  Ranges  oe  Conics.    The  Steinerian  Transeormation. 

CUBICS. 

§  45.  Pencils  and  ranges  of  conics 172 

§  46.  Products  of  pencils  and  ranges  of  conics 180 

§  47.  The  Steinerian  transformation 185 


CONTENTS.  vii 

PAGE 

§  48.  Curves  of  the  third  order 189 

§  49.  Curves  of  the  third  order  generated  by  involutoric  pencils 197 

§  50.  Various  methods  of  generating  a  circular  cubic 204 

§  51.  The  five  types  of  cubics  in  the  Steinerian  transformation 209 

CHAPTER  V. 

Applications  in  Mechanics. 

§  52.  A  problem  in  graphic  statics 217 

§  53.  Statical  proofs  of  some  projective  theorems 220 

Geometry  of  stresses  in  a  plane 223 

§  54.  General  remarks 223 

§  55.  Involution  of  conjugate  sections  and  stresses 224 

§  56.  Discussion  of  this  involution 227 

§  57.  The  stress  ellipse.     Metric  properties  of  the  involution  of  stresses  . .  229 

§  58.  Examples 236 

§  59.  The  rectangular  pair  of  the  involution  of  stresses  in  nature 239 

Realization  of  collineations  by  linkages 242 

§  60.  Introductory  remarks 242 

§  61.  Analytical  formulation  of  the  problem  . 244 

§  62.  Peaucellier's  inversor 246 

§  63.  Pantographs 248 

§  64.  Rotator  and  its  combinations 251 

§  65.  Translators 252 

§  66.  Linear  transformation 253 

§  67.  Perspective 257 

Index 261 


PROJECTIVE   GEOMETRY. 


CHAPTER  I. 


GENERAL  CONSIDERATIONS.  ANHARMONIC  RATIO.  PROJEC- 
TIVE RANGES  AND  PENCILS.  POLAR  INVOLUTION  OF  THE 
CIRCLE. 

§  I.  Geometric  Quantities  and  their  Signs. 

Geometric  quantities  can  be  represented  by  numbers  by 
assuming  an  arbitrary  geometric  quantity  of  the  same  kind  as  a 
unit.^  To  show  this  for  Hnear  quantities,  assume  any  Hne  AZ^ 
Fig.  I,  and  a  unit  u.    Measure  off  on  AZ  as  many  units  u  as 


Fig.  I. 

possible,  so  that  the  remainder  BZ<u.  Suppose  that  the  num- 
ber of  units  measured  on  ^Z  is  a,  so  that  AZ-=au-\-BZ.  Now 
consider  BZ  as  a  unit  and  u  as  the  quantity  to  be  measured. 
Suppose  that  BZ  is  contained  h  times  in  u  and  that  the  remain- 
der r<^BZ.  Then  u  =  h-BZ+r.  In  a  similar  manner,  consider  r 
as  a  unit  and  BZ  as  the  quantity  to  be  measured.  Suppose  that 
r  is  contained  c  times  in  BZ  and   that   the   remainder  is  s,  so 

^  See  Lagrange's  Lectures  on  Elementary  Mathematics  (translated  by  Th.  J. 
McCormack,  Open  Court  Publ.  Comp.,  Chicago),  p.  3. 


2  PROJECTIVE  GEOMETRY. 

that  BZ  =  cr-\-s.     Continuing   this    process   till  it  closes,  or  else 
indefinitely,  there  results  the  series 

AZ=au+BZ, 
u  =  b-BZ+r, 
BZ  =  cr-\-s, 

r  =  ds-\-t, 

s  =  et-\-v, 


which  by  elimination  leads  to  the  continued  fraction  (u  as  the 
original  unit  being  i) 

AZ=a+i 


b+i 


c+i 

d+i 

^.   .  . 

This    evidently    represents    a    number    and    the    proposition    is 
proved.  ^ 

Geometric  quantities  being  represented  by  numbers,  it  must  be 
possible  to  define  negative  and  positive  geometric  quantities  in 
accordance  with  the  laws  of  Arithmetic.  This  can  be  done  in  the 
most  convenient  manner  by  the  method  of  displacements.  As- 
sume any  line  and  three  points  on  it  in  the  order  A,  B,  C  from 

^  This  continued  fraction  is  convergent,  since  by  the  process  of  its  formation 
it  continually  approaches  the  limit,  which  is  known  in  advance.     For  example, 

^2=1+1 

2+1 

2+1 

2+     .    .    . 

Concerning  further  details  consult  Laurent:    Traite  d'Analyse,  Vol.  V,  pp.  321- 
359- 


GENERAL  CONSIDERATIONS.  3 

the  left  to  the  right.  Increase  and  decrease  of  geometric  quan- 
tity on  this  line  are  measured  by  the  amount  of  displacement  of  a 
moving  point,  or  by  the  length  of  the  hne  between  the  original 
and  final  position  of  a  moving  point  on  this  line.  The  formal 
laws  of  all  displacements  on  this  line  are  those  of  the  group.     Thus, 

(i)  AB-\-BC=-AC 

shows  that  two  displacements  succeeding  each  other  are  equiva- 
•lent  to  a  single  displacement  of  the  same  kind  and  of  the  same 
system  (group). ^     It  follows  further  that 

(2)  AB+BC+CA=o]'' 

hence  by  substitution  of  (i)  in  (2) 

AC+CA=o, 
or  CA  =  -AC\ 

i.e.,  two  displacements,  or  geometric  quantities,  which  are  de- 
scribed in  opposite  directions  are  of  opposite  sign.  The  same 
conclusions  are  reached  when  angular  displacements  are  con- 
sidered. It  is  a  universal  -convention  to  designate  all  geometrical 
quantities  which  are  obtained  by  displacements  on  a  line  from 
the  left  to  the  right  as  positive  and  those  in  the  opposite  direction 
as  negative.  In  a  similar  manner,  angles  formed  by  angular  dis- 
placements counter-clockwise  are  assumed  as  positive  and  those 
clockwise  as  negative.  The  conception  of  the  group  is  general 
and  also  comprises  the  determination  of  such  geometric  quan- 
tities as  areas  and  volumes. 

In  case  of  a  surface  assume  a  pole  O  and  any  line  I  on  this 
surface.  A  point  P  is  moving  on  I,  and  in  any  position  of  its 
motion  is  connected  to  the  point  O  by  a  geodesic  of  the  surface. 
The  generalized  radius  vector  OP  then  sweeps  over  a  certain  area 

^  It  is  beyond  the  Kmits  of  this  book  to  enter  into  a  discussion  of  groups  in 
this  connection. 

^MoBius:  Barycentrische  Calcul,  \  1. 


U' 


PROJECTIVE  GEOMETRY. 


which  is  subject  to  the  laws  of  the  group.      Thus  if  P  moves 
from  ^  to  -S  to  C,  counter-clockwise  with  respect  to  O, 

OAB+OBC  =  OAC, 
OAB+OBC+OCA=o; 


hence 


or 


OAC+OCA^o, 

OCA  =-OAC. 


Two  areas  have  therefore  to  be  considered  as  of  opposite  sign 
if  their  boundaries  are  described  in  oppo- 
site senses.  Accordmg  to  the  distinc- 
tion which  we  have  made,  an  area  is 
positive  or  negative  according  as  its 
boundary  is  described  counter-clockwise 
or  clockwise,  respectively.  From  this  it 
follows  further  that  if  the  point  P  de- 
scribes a  closed  line  on  a  surface,  the 
radius  vector  OP  sweeps  over  an  area 
equal  to  the  area  enclosed  by  the  curve. 
For  a  triangle  ABC,  Fig.  2, 

Fig.  2.  ABC  =  OAB+OBC+OCA, 

where  OCA  is  negative.     On  the  other  hand 
CBA = OCB + OB A  +  OAC 


hence 


=  -OBC-OAB+OAC; 
ABC + CBA  =0, 

CBA  = -ABC. 


The  same  reasoning  may  be  extended  to  the  determination  of 
volumes,  which  is  left  to  the  reader  as  an  exercise. 

Ex.  I.  li  A,  B,  C,  D  Sire  four  collinear  points,  prove  that 

BC-AD+CA-BD  +  AB-CD  =  o. 


GENERAL  CONSIDERATIONS.      ^  5 

Ex.  2.  For  tlie  same  points  prove 

DA^-BC+DB^-CA+DC^-AB  =  -BC-CA-AB. 

§  2.    Anharmonic    Ratio. ^      Projective    Transformation   of   the 
Points   of  a  Straight   Line. 

I.  Critical  Note. — von  Staudt  in  his  classical  works  ^  on  the 
geometry  of  position  created  a  system  with  the  principal  purpose 
of  laying  the  foundations  of  geometry  without  the  aid  of  metrical 
considerations.  He  introduced  the  word  "Wurf"  as  an  equiva- 
lent of  anharmonic  ratio  and  attached  to  it  a  meaning  independent 
of  any  ratio.  The  anharmonic  ratio  is  considered  as  a  property 
of  the  "  Wurf ",  so  that,  according  to  v.  Staudt,  metrical  geometry 
is  based  upon  projective  geometry,  or  rather  the  geometry  of 
position.  Steiner,  on  the  contrary,  took  the  anharmonic  ratio  as 
a  starting-point  in  his  investigations.^  In  a  recent  paper  *  Pom- 
CARE  has  pointed  out  "that  from  a  certain  point  of  view  the 
geometry  of  v.  Staudt  is  predominantly  a  visual  geometry,  while 
that  of  Euclid  is  predominantly  muscular."  In  other  words,  the 
two  geometries  are  derived  from  experiences  in  optics  and  kine- 
matics, respectively. 

In  works  with  practical  purposes,  where  applications  form  an 
important  part,  it  is  probably  of  the  greatest  advantage  to  take 
one  view  or  the  other  according  to  the  simplicity  of  the  treatment 
which  it  may  afford. 

This  method,  although  objectionable  from  the  standpoint  of 
pure  geometry,  reflects  the  development  of  geometric  science 
itself. 


'  I  shall  use  the  expression  anharmonic  ratio,  because  it  is  used  by  the  trans- 
lators of  Reye's  and  Cremona's  treatises  on  projective  geometry  and  by  a  majority 
of  English  authors.  Double  ratio,  corresponding  to  the  German  Doppelverhalt- 
nis,  is  presumably  a  better  designation. 

^  Geometrie  der  Lage,  1847.     Beitrdge,  1856-60. 

^  Systematische  Entwickelung  der  Abhdngigkeit  geometrischer  Gestalten,  etc., 
1832. 

*  On  the  Foundations  oj  Geometry,  Monist,  No.  i,  Vol.  IX. 


6  PROJECTIVE  GEOMETRY. 

2.  The  anharmonic  ratio  of  four  points  A,  B,  C,  D  on  d,  line, 


-^   d 


Fig.  3. 

a  straight  line  for  the  sake  of  simplicity,  where  {A,  B)  shall  be 
designated  as  the  first,  (C,  D)  as  the  second  pair,  Fig.  3,  is 


(I) 


AC    I  AD 


BC  J  BD 


As  AC,  BC,  AD,  BD  are  all  positive  quantities,  k  will  be  a  posi- 
tive number.  It  is  clear  that  this  is  not  the  only  anharmonic 
ratio  that  may  be  formed  between  the  four  points.  As  there  are 
24  permutations  possible  between  four  elements,  there  will  also 
be  24  anharmonic  ratios.  Some  of  these,  however,  have  the 
same  value,  and  it  may  easily  be  verified  that  there  are  only  6 
different  anharmonic  ratios  possible.  Designating  (i)  by  the 
symbol  (ABCD),^  these  are 


(2) 


f  (ABCD)  =  k, 
(BACD)=j, 

(BCAD)^^, 

b 
(CBAD)  = 


fA'^)h(^)T  k-J: 


-k' 


(CABD)  =  ^~^, 
{ACBD)  =  i-k. 


If  the  points  A,  B,  C,  D  are  located  by  their  displacements  a,  b, 


^  MoBius,  Barycentrische  Calcul,  §  183. 


(Vt 


Va^^ 


PROJECTIVE   TRANSFORMATION .  7 

c,  d  from  a  fixed  point  O,  the  first  anharmonic  ratio  assumes  the 
form 

c—a    id— a      ,  , 

(3)  .  7^/dZi-k.  '    -     ^  ) 

3.  This  expression  leads  to  the  solution  of  the  important  prob- 
lem to.J^d  all  pairs  of  points,  X,  Y,  which  with  two  fixed  points 
A  and  B  form  the  constant  anharmonic  ratio  k.  Associating  with 
X  and  Y  the  displacements  x  and  y  from  O,  the  condition,  ac- 
cording to  (3),  is 

x-a        y-a 

(4)  Xr-r  =  ^; 


or,  solved  for  x, 
(5) 


x—b        y—h ' 

\ 
(a—bk)y—ab(i  —  k)  ] 

{i-k)y-{b-ak)  '  I 


From  this  it  is  seen  that  to  every  value  of  y  corresponds  one 
and  only  one  value  of  x  satisfying  the  condition  of  the  problem, 
and  vice  versa.     Taking  any  four  points  Y^,  Fj,  F3,  F4,  and  de- 
termining the  corresponding  points  X^,  X2,  X^,  X^  according  to  ' 
(5),  there,  is  found  the  relation^  ^^  (^  (^^^^  ^  ' 

(x.x,x,x,)==(Y,Y,Y,Y,y, '  ^^-^^  ^  .  ^ ^^;;;;'^ 

i.e.,  any  four  points  of  the  series  (X)  and  their  corresponding 
points  of  the  series  (F)  satisfying  the  condition  (5)  have  the  same 
anharmonic  ratio.  Two  series  or  ranges  of  points  with  ,this 
property  are  said  to  be  projective.  Formula  (5)  is  the  analytical 
expression  for  these  projective  ranges  of  points;  it  effects  a  pro- 
jective transformation  ^  of  the  points  of  a  straight  line. 

For~~y~^^d^  =  a,  and  for  y  =  b,  x=b;  i.e.,  the  transformation 
leaves  the  points  A  and  B  invariant;   they  are  called  the  double- 

^  The  word  projective  was  first  used  by  Poncelet  in  his  great  work :  Traite 
des  proprietes  projectives  des  figures,  1822.  MoBius  was  the  first  who  gave  an 
analytical  representation  of  projective  transformations,  in  Der  barycentrische 
Calcul,  1827. 


PROJECTIVE  GEOMETRY. 


points  of  the  transformation,  or  of  the  projective  ranges  of  points. 
From  (4)  follows  immediately  that  every  pair  of  corresponding 
I  points  forms  a  constant  anharmonic  ratio  with  the  double- points. 
On  the  other  hand  every  transformation  of  the  form 


(6) 


x= 


Ay+B 
Cy+D 


is  projective.  To  prove  this  assume  four  points  F^,  Y^,  F3,  Y^^ 
and  determine  the  corresponding  points  X^,  X,,  X3,  X4.  Let 
Jii  y-ii  y^i  Ji  ^^^  ^15  ^2)  ^3j  ^4  be  the  corresponding  displacements, 
then  to  form  {X^X^X^X^  we  have  from  (6) 

{AD-BC){y,-y,) 


(7) 


VCa        -^i 


t/V  A  x\") 


(Cy,+D)(Cy,+D)' 

JAD-BC){y,~y,) 
'     {Cy,+D){Cy,+D)^ 

(AD-BC)(y,-~y,) 
{Cy,+  D)[Cy,+D)' 

(AD~BC)(y-y,) 
{Cy,+  D){Cy,+Dy 


and  by  division 


1  /x^-x^^y^-y,  ly^-y^ 

./   X.—  X^       V,— Vo/    -v.  — 'Vo' 


or 


X3    x^'  x^    X2    y^    y2    yi    y^ 

{x,x,x,x:)={y,y,y.j:). 


which  is  a  property  of  a  projective  transformation.  To  prove 
that  (6)  is  of  the  form  (5),  we  find  the  double-points  of  (6)  by 
putting  y  =  x\  then  (6)  becomes 

(8)  Cx'-{A-D)x-B  =  o\ 

hence,  designating  the  roots  of  this  equation  by  a  and  &, 

A-'D-^\/{A-Dy^^BC 


(9) 


a  = 


2C 


b  = 


A-D-V(A-Dy+4BC 


V  '   l>tiU//L/i- 


2C 


M- 


fSwn^ 


AXk^ 


PROJECTIVE   TRANSFORMATION.  9 

The  transformation  (6)  has  therefore  two  double-points.  Put- 
ting in  the  first  and  third  equations  of  (7)  x^  =  y^  =  a  and 
Xi=yi=h,  it  is  found  by  division  that 

a-x,  ih-x^     Cb  +  D 
(^°)  ^^7/  b^rC^+D  =  ^  (constant). 

Thus  we  find  that  any  pair  of  corresponding  points  of  the 
transformation  (6)  forms  a  constant  anharmonic  ratio  with  its 
double-points;  such  a  transformation  is  projective.  In  deriving 
equation  (5)  it  was  assumed  that  A  and  B  are  real  points. 
Assuming  a  projective  transformation  of  the  form  (6),  where 
A,  B,  C,  D  are  real  coefficients,  it  may  happen  that  the  double- 
points  given  by  (9)  are  imaginary.  In  fact  there  are  three  pos- 
sibilities for  the  double  points.     According  as 


(II)  {A~Dy+^BC^o, 


a  and  h,  or  the  double-points,  are  real,  real  and  coincident,  or 
imaginary,  and  the  transformations  are  then  called  hyperbolic, 
parabolic,  or  elliptic. 

4.  We  shall  next  show  that  two  projective  ranges  are  deter- 
mined by  three  pairs  of  corresponding  points  X^,  Y^ ;  Xg,  F2 ;  X^,  F3, 
whose  positions  are  determined  by  the  coordinates  x^,  y^ ;  X2,  ^'25  •  •  • 
If  these  points  are  corresponding  in  two  projective  ranges,  their 
coordinates  must  satisfy  some  relation  of  the  form 

ay+b 
cy+d 

or  cxy+dx—ay—b  =  o. 

c    d   a         .  .  .       . 

To  determine  the  ratios  7-,  7-,  7-,  vi^hich  evidently  determine  the 

transformation,  we  have  the  conditions 


lO 


PROJECTIVE  GEOMETRY. 


c  da 

c  d        a 

c  da 

These  are  three  equations  with  the  three  required  ratios  as 
unknown  quantities.  These  are  therefore  uniformly  deter- 
mined by  the  x's  and  y's  and  are  in  determinant  form: 


I 

X, 

yi 

I 

vVo 

y2 

I 

^3 

Jz 

x,y. 

X, 

Jl 

^2)'2 

X2 

72 

^3}'3 

Xs 

y^ 

x,y. 

I 

yi 

^23'2 

I 

y2 

^373 

I 

ys 

x^yx 

X, 

yi 

x^y-i 

X2 

y2 

Xsys 

X3 

ys 

x,y. 

-X, 

I 

X2y2 

-X2 

I 

x^y^ 

~x. 

^ 

x{yx 

X, 

yi 

^2^2 

X2 

72 

x^y^ 

Xs 

3'3 

4 


This  proposition  is  also  geometrically  clear.  In  two  pro- 
jective ranges  any  four  points  of  one  range  have  the  same  anhar- 
nlonic  ratio  as  the  four  corresponding  points  of  the  other  range. 
Hence,  choosing  any  fourth  point  x^,  then  there  is  clearly  only 
one  point  Y^,  so  that 

{X,X2X,X,)  =  {Yj2yzY,y, 

i.e.,  three  pairs  of  points  determine  the  projectivity. 

As  an  exercise  assume  the  case  of  two  coincident  projective 
ranges  for  which  the  infinitely  distant  point  is  self-correspond- 
ing.     Let  x^  =  3'i  determine  this  infinitely  distant  point.     From 

c  ,      d  a 

the  above  expressions  we  find  r  =  o,  while  t"   and  -r  are  finite. 

The  projective  transformation  assumes  the  form 

x=ay-{-^; 
i.e.,  what  is  called  a  linear  transformation. 


INVOLUTION.  II 

5.  It  is  beyond  the  limit  of  this  book  to  discuss  all  special 
cases  of  projective  transformations  of  a  straight  line  in  detail. 
We  shall  indicate  one  of  its  properties  which  is  of  extreme  impor- 
tance in  modern  geometry,  and  then  discuss  the  special  case  of 
involution.  Let  a  point  x  be  transformed  into  a  point  x'  by 
the  projective  transformation 

(12)  x^  = ^. 

^     ^  yx^  0 

Transform  x/  into  a  point  x"  by  another  transformation  of 
the  same  kind: 

(13)  •  ^  = — rr^- 

The  result  of  these  two  operations  is 

(a:c^,  +  rA):v+ (/?«,  + o^A) 

which  shows  that  xf'  is  obtained  from  x  by  a  projective  trans- 
formation of  the  form  (12).  Hence  one,  two,  or  more  opera- 
tions of  the  form  (12)  in  succession  are  equivalent  to  an  oper- 
ation of  the  same  kind.     Giving  a,  /?,  y,  d  all  possible  real  values, 

<x     B     T 
(12)  depends  upon  the  three  ratios—,  — ,  -^,  so  that   (12)  repre- 
sents   a    triply    infinite    number    of    projective    transformations. 
For  this  reason  it  is  said  that  all  projective  transformations  0}  a 
straight  line  form  a  continuous  three-termed  group  (dreigliedrig) .^ 

§  3.  Involution.^ 

In  case  that  the  constant  anharmonic  ratio  k  in  equation  (4) 
of  the  foregoing  paragraph  is  —  i, 

^  SOPHTJS  Lie:    Vorlesungen  iiber  continuierliche  Gruppen. 

^  First  systematically  studied  by  Desargues  (Brouillon  projet,  etc.). 


12  PROJECTIVE  GEOMETRY. 

x—a        y—a 
^  ^  x—b        y—o 

(a+h)y—  2ah 
^  2;y— (a+o) 

In  these  equations  x  and  ;y  can  be  interchanged  without  affect- 
ing (i)  or  (2).  The  ratio  {ABXY)  =  —  i  is  called  a  harmonic  ratio 
and  (i)  and  (2)  represent  an  involutoric  transformation.     To  the 

a+b 
point    at   infinity,    y=oo,    corresponds   the    point  x  =  - •;  i.e., 

the  point  bisecting  the  distance  AB  between  the  double-points. 
It  is  called  the  middle  point  of  the  involution.  Designating  this 
point  by  M,  it  is  found  that      k     «-  ,  ^  '      6ei«'^)-'/<«-^ 

-(2)  MX-MY=~ -= ;■-    ^1^^    z^ 

i.e.,  the  product  0}  the  displacements  0}  two  corresponding  points  of 
an  involution  from  the  middle  point  is  constant  and  equals  the 
square  of  the  displacement  of  either  double-point  from  the  middle 
point. 

Equation  (2)  may  always  be  written  in  the  form 

{4)  x=^-^, 

If  a  and  b  are  given  by  the  values 

y      ^Y        y  X      ^  J        y 

As  these  expressions  define  the  double-points,  they  m.ust  also 
result  directly  from  (4).  For  the  double-points  x=y\  hence 
from  (4) 

yx^—  2ax-\r^  =  o. 


INVOLUTION.  13 

The  roots  of  this  equation  are  indeed  identical  with  the  pre- 
vious values  of  a  and  b.     li->^,a  and  b  are  conjugate  complex 

numbers,  i.e.,  the  double-points  of  the  involution  are  imaginary. 
In  this  case  the  middle  point  M  of  the  mvolution  is  still  real, 

a-\-b      a         , 

smce =  -  ,  and 

2  r 

MX-MY= ^=— -^<o; 

4         r      r 

B     a2 
X  and  Y  are  on  different  sides  of  M.     For  -  =  — 5  the  double- 

points  coincide,  and  MX-MY=o;  every  point  corresponds  to  M.-V  (~tj 
•  According  to  these  results  involution  has  been  classified  as  hyper- 


v.:  / 


AJukhjl,  '<5^ 


i;i.  ^  )y\\ 


Fig.  4. 


bolic  in  case  of  real  double-points,  elliptic  in  case  of  imaginary 
double-points,    parabolic    in    case    of    coinciding    double-points. 


I-- 


<y 


)(^ 


Fig.  5. 


Geometrically,  the  different  cases  may  be  obtained  as  intersec- 
tions of  a  straight  line  with  coaxial  systems  of  circles.     Figs.  4, 


14 


PROJECTIVE  GEOMETRY. 


5,  and  6  represent  hyperbolic,  elliptic,  and  parabolic  involutions 
respectively.  In  the  first  the  points  of  a  pair,  XY,  are  always  on 
the  same  side  of  M,  and  move  in  opposite  directions;  in  the  second 
they  are  on  different  sides  of  M,  and  one  of  the  points  {X)  is 


Fig.  6. 


within  the  distance  AB  and  the  other  without.  Corresponding 
points  move  in  the  same  direction. .  We  have  seen  that  an  in- 
volution on  a  straight  line  is  determined  by  the  transformation 


(5) 
(6) 


X  =■ 


ax-\-h 


ex— a 


or 


-(t; 


cxx'  —  a(x+  x')  —  h  =  o, 


which  shows  that  an  involution  is  determined  by  two  pairs,  since 
there  are  only  two  essential  constants  in  (6). 
Suppose  that  in  a  projective  transformation 


ax+b 
x-=- — —.,     or 
cx^d 


cxx'-\-dx/—ax—b  =  Of 

the  points  x/,  x^  may  be  interchanged  without  affecting  the  pro- 
jectivity.     The  condition  for  this  is 


(7) 
(8) 


cx^x^  +  dxi'—  ax^—  b=o, 
cx^x/ + dx^  —  ax/  —b  =  o. 


PENCILS  AND   RANGES.  1 5 

By  subtraction 

(9)  6i!(:Vj— x/)  +  a(Xi— x/)=o, 

which  can  only  be  satisfied  when  d=  —  a,  since  x^^t^x/.  The 
condition  d=  —  a,  however,  impHes  involution,  hence  the  theorem: 
//  a  projective  trans forniaiion  contains  a  pair  whose  values  may 
he  interchanged  without  altering  the  transformation,  it  is  an  involu- 
tion.    Thus  if  XiXj^  be  a  pair, 

(10)  (ABX,X,')=ABX/X,)  =  - 1. 

§  4.  Projective  Pencils  of  Rays. 

Let  a,  b,  c,  d  be  four  rays  (straight  lines)  passing  through 
a  common  point,  and  (ab),  (be),  etc.,  the  angles  included  by 
the  rays  a  and  b,  b  and  c,  etc.,  so  that  also  here  (ab)  =  —  {ba); 
(ab)+(bc)-\-(ca)  =  o. 

In  analogy  with  the  anharmonic  ratio  of  four  points,  the  anhar- 
monic  ratio_ofJhe.se_ia^  is 


sin  (ac)   /sin  (ad) 


sm  {ac)  I 
^^'  sin  ibc^  I 


=  k. 


sin  {be)  I   sin  {bd) 

and  may  be  designated  by  {abed)  =  k. 

What  has  been  said  about  the  permutations  of  four  points 

'   ■•'  applies  without  alteration  to  four  rays.     Consider  now  four  con- 

current rays  a,  b,  c,  d  passing  through  four  points  A,  B,  C,  D  oi 

"^  a  straight  line,  respectively.     From  Fig.  7  it  is  seen  that 


sm  (be)/  sin  (bd)~  CP  /  DO' 


sin  (ac)  /sin  (ad)     CN  /DM 


/sm  (ad)     Cl\    / 
/sin  (bd)~~Cp/  ' 


1' 


K-,,,        DM  and  CN  being  ±  a  and  DO  and  CP  ±  to  b.     But 

CN     AC  ,      CP     BC     , 

7777=  ~7^     and     iR7^  =  ^rF:;    hence 
^'      -11  A         DM     AD  DO    BD' 

CN  /DM    AC  /AD 
CPI  DO~Bcl  BD 

(2)  (abcd)  =  (ABCD). 


i6 


PROJECTIVE   GEOMETRY. 


This  important  result  may  be  stated  by  the  theorem: 

The  anharmonic  ratio  of  any  four  concurrent  rays  is  equal  to 

the  anharmonic  ratio  of  four  points  formed  by  the  intersection  oj 

any  transversal  with  these  rays.     (Pappus.) 


Fig. 


In  other  words,  if  the  rays  a,  b,  c,  d  are  considered  as  project- 
.  ing  rays  in  a  central  projection,  such  a  projection  does  not  change 
the  anharmonic  ratio  of  four  points. 

A  system  of  rays  in  a  plane  and  passing  through  the  same 
point  is  called  a  pencil  of  rays.  ^  By  the  above  theorem  all  proper- 
ties of  projective  ranges  of  points  may  be  transferred  to  pro- 
_J.ective  pencils^ of  rays. 

In  order  to  obtain  an  analytic  expression  for  the  rays  of  two 
projective  pencils  with  the  same  vertices,  assume  the  Hne  repre- 
senting a  projective  range  of  points  as  the  A'- axis  and  the  origin 
of  the  range  as  the  origin  of  a  Cartesian  system.  Let  F,  with 
the  coordinates  m  and  n,  be  the  center  of  a  pencil,  then  the  equa- 
tions of  the  rays  passing  through  the  double  points  A  and  B  of  the 
transformation 


(3) 


x  =  - 


{a—bk)y—ab{i  —  k) 
{i  —  k)y—  (b—ak) 


(eq.  S,  §  2) 


^  Cremona,  loc.  cit.,  p.  22.  In  the  translation  of  Reye's  Geometric  der  Lage 
the  term  "sheaf  of  rays"  is  used,  while  in  Cremona's  treatise  "sheaf  of  rays" 
or  "planes"  means  all  rays  or  planes  passing  through  a  point  in  space.  Gar. 
Strahlenhuschel .     Fr.  Faisceaux. 


are 
(4) 

(5) 


PENCILS  AND  RANGES. 

nx+  {a—  m)y—  aw =o, 
nx-\-  {b—m)y—bn=o. 


17 


/(^, 


Multiplying  (5)  by  X  and  subtracting  from  (4),  the  equation 
of  a  third  ray  through  V  results: 


^         .(6), 


nx+ 


a—Xh 


a—  Xb 


m  ]'v— 


-n  =  o. 


,i-A     "7''      I- A 

This  ray  intersects  the  A'-axis  in  a  point,  say  Z>,  whose  abscissa 
To  find  the  corresponding  point  C  in  transformation 


d= 


i-A 


a—  Xb 


(3),   put  y  =  d  = r  in  (3).      This  gives  for  the  abscissa  c  of  C 

I  —  A 

a—Xbk^^  .         ,    . 

;  SO  that  the  equation  of  the  ray  passmg 


the  value    c  =  ,, 

i  —  Xk 

through  C  becomes 


(7) 


nx+ 


a—Xbk 
i-Xk' 


■m  jy 


a—  Xbk 
i-Xk 


n  =  o. 


Comparing  equations  (6)  and  (7)  with  those  of  (4)  and  (5), 
we  find,  if  (4)  and  (5)  are  wTitten  u  =  o,  v  =  o,  that  (6)  and  (7), 
the  equations  of  the  rays  VD  and  VC,  are 


(8) 
(9) 


w—  Xv  =0, 
u—  Xkv=o. 


For  each  value  of  X  these  equations  represent  a  corresponding 
pair  in  a  projective  transformation  of  rays  which  is  characterized 
by  the  anharmonic  ratio  k.  In  other  words,  for  a  variable  •  A, 
(8)  and  (9)  represent  two  projective  pencils.   \a.  5.  w  X  -+•  (a--\C)u -<Ji>^  ~  D 


ouC 


l8  PROJECTIVE   GEOMETRY. 

§  5.  Involutoric  Pencils. 

In  the  case  of  involution  the  anharmonic  ratio  is  ^=  —  i,  so 
that  equations  (8)  and  (9)  of  the  previous  paragraph  become 

(i)  u—ku=o, 

(2)  u-{-Xv=o\ 

i.e.,  if  u  and  v  are  any  two  rays,  the  rays  u—Xv  =  o  and  u-{-Xv=o 
are  harmonic  with  regard  to  u  and  v.  For  X  =  o  and  /^=oo  the 
double-rays  w  =  o  and  v  =  o  oi  the  involution  are  obtained,  (i) 
and  (2)  define  an  involution  of  rays  when  A  varies  from  —  00  to 
+  00 .     Suppose 

u  =  ax  -\-hy  -\-c  =0, 

'J  v  =  a^x-^h^yArC^^o 

be  the  equations  of  the  double-rays,  so  that  (i)  and  (2)  assume 
the  form 

(3)  {a—  Xa^x-]-  (b—  kh^y^c—  Xc^  =  o, 
"^^^^      (4)  {a+Xa,)x+{h^Xh;)y-Vc^Xc,-=o. 

The  trigonometric  tangents  of  the  angles  of  inclination  with 
+  X,  or  the  slopes  of  (3)  and  (4),  are 

a—  Xa. 

a+Xa. 

hence  the  tangent  of  the  angle  4>  included  by  (i)  and  (2)  or  (3) 
and  (4)  is 

(7)  tan^-^  ^   -         '  ^ 


I  +  W-W,       02+&2_P(a^2+&^2)* 


PENCILS  AND   RANGES.  19 

This  shows  again  that  for  A  =  o  and  A  =  00 ,  tan  ^  =  o,  or  0  =  o 
(180°).  In  these  cases  the  rays  (3)  and  (4)  coincide  and  the 
double-rays  of  the  involution  are  obtained.  Supposing  that 
a-p—  ab^^o,  which  generally  will  be  the  case,  we  may  ask  for 

those  values  of  X  which  will  make  tan  0=oo,  or  ^=7,  a  right 

angle.     From  (7)  we  find  for  this  condition 

a2+&2-P(ai2+V)=o,    or    . 


(8) 


which  is  always  a  real  quantity.  Whether  we  take  the  +  or  — 
sign  for  A  in  (8),  we  obtain  the  same  couple  of  equations  (3)  and 
(4);  hence  the  theorem:  .; 

An  involution  of  rays  always  contains  one,  hut  only  one,  rect-  \ 
angular  pair.  '  \ 

We  shall  now  discuss  the  case  where  tan  ^  =  00,  or  ^  =  90°, 
for  all  values  of  A.  In  order  that  this  be  the  case,  the  quanti- 
ties a,  h,  a^,  &i  must  satisfy  the  conditions  a^-\-b^  =  o,  a^^+bj^  =  o, 
afi—abi  y£o,  or  b  =  ±ia,  &i=  ^  ia^,  so  that  the  equations 

■'  '  ■   "^i-r  - 
ax-\-by-\-c=o,     aiX-\-b^-\-c^=o 

of  the  double-rays  assume  the  forms 

c  .      c. 

x-\-iy-\--=o,     x—ty-\ — =0, 


c         c 

and  are  imaginary.     We  can  dispose  of  the  constants  —  and  —  in 

such  a  manner  that  the  double-rays  will  pass  through  the  real 
point  (a,  /?).    Their  equations  then  become 


a^-h^fi 


^^^  \v  =  X 


^iX-\-iy—{pL-\-  i/?)  =  o, 
=  X—  iy—  {a—  i^)  =  o. 


A.  ^        0,1  . 


20  PROJECTIVE   GEOMETRY. 

The  involution  with   these  double-rays  has  only  rectangular 
pairs.     The  equations  of  such  a  pair  are 

\u-Xv  =  o, 

^    ^  \u-\-kv=o. 

For  real  values  of  X  the  pairs  are  imaginary,  since  (lo)  may 
be  written 


(II) 


J—X  .  i  +  ifJ- 

Puttmg  *~xi  =  /«)  a  real  quantity,  /=    _  .  .     Thus,  if  in  (lo)  we 

give  X  all  imaginary  values  contained  in  the  formula  X  — r-, 

I  ujJ. 

where  p.  is  any  real  quantity,  the  corresponding  pairs  (ii)  in  the 
involution  will  be  real  and  rectangular.  Now  an  involution  of 
rays  has  generally  only  one  rectangular  pair  and  is  determined 
by  two  pairs,  hence  the  theorem: 

(       An  involution  of  rays  having  more  than  one  rectangular  pair 
'         •  has  all  its  pairs  rectangular. 

"^  The  double-rays  of  this  involution  are  hnaginajy  and  pass 

"-^"^Z    through  the  two  infinite  points,  which,  as  will  be  seen  later  on, 
'"     are  called  the  circular  points  at  infinity,  §  12. 

If  an  involution  of  rays  shall  contain  the  rays  joining  the 
vertex  with  the  circular  points,  i.e.,  the  two  rays  with  the  slopes 
+  i  and  —  i  as  a  pair,  then  according  to  (3)  and  (4)  we  must 
have  ,   s^ 

iVr\^-l         ^'  ^~ ^^1      ,  •         a+_Xa^__. 

^^-.,  -  b-Xb,     "^*'        b+Xb~     ^' 


^ATvilA)!^ 


or  X(bii—aj)-}-a—bi  =  o, 

X(bii-\-ai)  +  a-j-bi  =  o. 


\ 


PENCILS  AND    RANGES.  21 

These  two  equations  must  exist  for  the  particular  value  of 
A  which  makes  the  slopes  of  (3)  and  (4)  +i  and  —i.  This  can 
only  be  true  under  the  condition 

or  aa^=~bb^.     O1.    -^  -       -r-  .   U 


/       )T\    i    ~    -i-  -^   KA^Ix^J^r^      j 


Hence  the  theorem: 

If  an  involution  of  rays  contains  the  rays  with  the  slopes  +i 
and  —i  as  a  pair,  then  the  double-rays  of  this  involution  are  per- 
pendicular to  each  other. 

Conversely,  it  can  easily  be  proved  that  if  the  double-rays  of 

an    involution   are   perpendicular,    then   this   involution   contains 

the  rays  with  the  slopes  +i  and  —i  as  a  pair.     The  slopes  of 

the  rays  of  any  pair  in  an  involution,  as  defined  by  (3)  and  (4), 

a—Xa^        ,       «+>^«i       ^  ,      ,  ,     , 

are  —  t — jr   and   —  ,      .,,  .      Consequently  the    tangents  of   the 

angles  which  these  rays  make  with  one  of   the  double- rays,  for 


7 


instance  w  =  o  (slope— 7-), 


are 


a     a—  Xa. 

-r  + 


b      b—Xb,  X(abi—ajj) 

a  a—  Xa^ 
b  b-Xb, 


a  a—Xa^     a-+b^— X(aa^-\-bb^) 


and 


a     a-\- Xa^ 

b      b+Xb^   __        X{ab^—aJ)) 


a   a-^Xa^     a'^+b''"-\-X{aa^+bb^)' 
^'^b'b+Xb, 

In  case  of  perpendicular  double-rays  aai-\~bbi  =  o,  and  these 
two  tangents  become  equal.     Hence  the  theorem: 

In  case  of  perpendicular  double-rays,  the  angles  of  all  pairs 
0}  the  involution  are  bisected   by  the  double-rays.      In  such  an  |   La-^'-^- 
involution  two  rays  chosen  from  each  of  two  pairs  include  the  same  ] 
angle  as  the  remaining  two  rays  of  the  two  pairs.  j 


J^..   Cy 


V  ^  .'"/l^^ 


'       ^2C^^ 


22  PROJECTIVE   GEOMETRY. 


§  6.   Product  of  Projective  Pencils  and  Ranges.* 

I.  In  §  4  it  has  been  shown  that  the  equations  of  two  pro- 
jective pencils  of  rays  with  the  same  vertex  may  always  be  writ- 
ten in  the  form 

(i)  u—  Xv  =  o, 

(2)  u—Xkv  =  o. 

For  every  value  of  X  these  equations  represent  a  correspond- 
ing pair  of  rays  in  a  projective  transformation  which  is  charac- 
terized by  the  anharmonic  ratio  k.  In  other  words,  for  a  variable 
X  (i)  and  (2)  represent  two  projective  pencils.  Now  the  second 
pencil  (2)  may  be  moved  into  any  other  part  of  the  plane  with- 
out ceasing  to  be  projective  with  regard  to  (i).  This  operation 
does  evidently  not  change  the  general  form  of  (2);  only  the 
expressions  u  and  v  are  transformed  into  new  expressions  r  and 
s.  These  represent  two  rays  intersecting  each  other  in  the  ver- 
tex of  the  moved  pencil.  Thus  the  equations  of  the  two  pencils 
are 

(3)  u-Xv  =  o, 

(4)  r—Xks  =  o. 

For  each  value  of  X  there  are  two  rays  which  intersect  each 
other  in  a  certain  point  P.  If  X  successively  assumes  all  values 
between  —  00  and  +<»,-?  describes  a  locus  whose  equation  is 
obtained  by  eliminating  X  between  (3)   and  (4).     This  gives 

(5)  vr—kus=o, 

^  The  conception  of  pencils  of  curves  and  surfaces  represented  by  equations 
of  the  form  P+  XQ  =  o  is  due  to  Lame,  who  introduced  it  in  his  article,  Stir  les 
intersections  des  lignes  et  des  surfaces,  Gergonne's  Annales,  Vol.  VII,  1816-17,  pp. 
229-240.  The  generation  of  conies  by  projective  pencils  and  ranges  is  due  to 
Steiner  (Systematische  Entwickelung,  etc.),  who  called  it  his  "steam-engine" 
(Dampfmaschine) . 


PENCILS  AND   RANGES.  23 

an  equation  of  the  second  degree  in  x  and  y,  since  u,  v,  r,  s  are 
linear  in  x  and  y.     Hence  the  theorem: 

The  product  of  two  projective  pencils  of  rays  with  separate 
vertices  is  a  curve  of  the  second  order. 

2.  Each  linear  expression,  like  ax+by+c  =  o,  depends  on 
two  independent  coefificients,  so  that  the  equation  vr—  kus  =  o 
contains  eight  independent  coefficients.  Arranging  in  (5)  the  terms 
according  to  powers  of  x  and  y,  an  equation  of  the  form 

(6)  ax^+2bxy-\-cy^+2dx-]-2ey-{-f  =  o 

is  obtained,  where  a,  b,  c,  d,  e,  f  are  expressed  in  terms  of  the 
coefficients  of  w  =  o,  v  =  o,  r  =  o,  s  =  o. 

As  (6)  contains  only  five  independent  coefficients,  it  is  clear 
that  the  eight  coefficients  in  m  =  o,  v  =  o,  r  =  o,  5  =  0  may  always 
be  selected  in  such  a  manner  that  (5)  becomes  identical  with 
any  equation  of  the  form  (6).     See  problem  11  in  §  7. 

Hence  the  theorem: 

Every  curve  of  the  second  order  may  be  considered  as  the  prod-  | 
uci  of  two  projective  pencils  of  rays.  | 


Fig.  8. 

ts  (t) 

(5)  is  satisfied  by  u  =  v=o  and  r=s=o,  also  by  w  =  r=o 
and  by  v  =  s  =  o;  i.e.,  the  curve  of  the  second  order  passes 
through  the  vertices  of  the  projective  pencils  and  also  through 


24  PROJECTIVE  GEOMETRY. 

the  points  of  intersection  of  the  rays  v,  s  and  u,  r,  Fig.  8.     If, 

therefore,  we  want  to  write  (6)  in  the  form  (5),  we  have  to  choose 

two  points  5  and  T  on  the  given  curve  (6).     Suppose  that  u  =  o 

and  v  =  o  contain  S,  and  r=o  and  s  =  o  T,  then  u,  v,  r,  s,  each 

only  depend  on  one  coefficient  (slope),  so  that  (5)  depends  on 

these  four  coefficients  and  k,  which  makes .  five  coefficients  in 

all.     These   coefficients  may  therefore  be  uniquely  determined 

so  that  (5)  represents  or  is  identical  with  (6).     Now  two  points 

00  (— 1)00  a/. 

6"  and  T  may  be  chosen  in  =  00  ^  different  rays  on  a 

curve.  Each  of  these  determines  a  different  but  unique  form 
of  (5).  The  previous  theorem  may  therefore  be  stated  as  fol- 
lows: 

Every  curve  of  the  second  order  may  be  produced  in  a  doubly 
infinite  number  0}  ways  by  two  projective  pencils. 

At  the  same  time  we  have  proved  the  theorem: 

If  two  fixed  points  S  and  T  be  joined  to  a  point  P  which 
describes  a  curve  of  the  second  order  through  S  and  T,  the  pencils 
(SP)  and  (TP)  about  S  and  T  as  vertices  are  projective. 

3.  As  the  general  equation  of  a  curve  of  the  second  degree 
depends  upon  five  independent  constants,  it  is  clear  that  five 
points  of  the  curve  determine  it.  Designating  the  coordinates 
of  one  of  these  points  by  Xi,  yi,  there  is 

axi^+  2bxiyi+  cyi^+  2dxi+  2eyi+f  =  o, 
^'=1,  2,  3,  4,  5. 

These  are  five  equations  with  five  unknown  quantities 
-J,  'J,  ~r,  "y>  ~T,  which    may  be    found   by  the    usual   method. 

Hence  the  theorem: 

A  curve  of  the  second  order  is  determined  by  five  points. 

In  a  similar  manner  it  may  be  proved  that  two  projective 
ranges  of  points  can  be  represented  by  the  equations 

(7)  a-/z/?  =  o, 

(S)  .  ■jr-fiKd  =  0, 


PENCILS  AND   RANGES.  25 

where  a,  /?,  f^  ^  ^^^  ^^^  line-equations  ^  of  four  points  in  a  plane. 
Assuming  the  knowledge  of  line-coordinates,  the  proof  may 
be  made  without  difficulty  and  may  be  left  to  the  reader. 

For  every  value  of  [x  there  are  two  points  which  determine 
a  straight  line.  If  [i  successively  assumes  all  real  values,  this 
line  envelops  a  curve  whose  equation  is  obtained  by  eliminating 
a  from   (7)   and   (8)   and  which  is 

(9)  ^y—  Kad  =  o. 

This  is  an  equation  of  the  second  degree  in  line-coordinates 
and  consequently  represents  a  curve  of  the  second  class.  ^  In 
analogy  with  the  previous  statements  we  have  also  the  theorems: 

Every  curve  0}  the  second  class  may  be  produced  in  a  doubly 
infinite  number  of  ways  by  two  projective  ranges. 

If  two  fixed  tangents  S  and  T  be  intersected  by  a  line  P 
which  envelops  a  curve  of  the  second-class  tangent  to  S  and  T, 
the  ranges  (SP)  ajid  (TP)  on  S  and  T  as  bases  are  projective. 

A  curve  of  the  second  class  is  determined  by  five  tangents. 


§  7.  Exercises  and  Problems. 

I.  Assuming    (ABCD)=k,    fmd    the    values    of    the    other 
twenty-three  ratios  which  may  be  formed  with  the  four  points 
ABCD. 
\    2.  Do  the  same  when  (ABCD)  =  -\-i,  —  i. 

3.  If  X^,  X2,  X3,  X4  and   Fi,  Fj,  F3,  F4  are  corresponding 
points  in  a  projective  transformation,  verify  the  relation 

(X,X,X,X,)=^iYJ,Y,Y,)     by  using 

ax-\-b 


y- 


cx-{-d 


^  Line-coordinates  of  a  line  are  the  negative  reciprocal  intercepts  of  this  line 
with  the  coordinate  axes.  The  reader  is  referred  to  Salmon-Fiedler:  Analytische 
Geometrie  der  KegelschniUe,  6.  ed.,  Vol.  I,  pp.  120—128. 

^  This  statement  stands  for  a  definition. 


26  PROJECTIVE   GEOMETRY. 

''  4.  If  the  double-points  of  an  involution  are  a=o  and  &=oo, 
prove  that  the  in volutoric.  transformation  has  the  form  x-{-y^c. 
^  S.  li  X  and  y  are  a  pair  in  an  involution  with  the  double- 
points  a  and  h,  prove  the  relation 

{x—  a) {y—  x) -\-  {x—  h){y—x)-\-2 {x—  a) {x—  h)  =  o. 

yj  6.  Also  establish  the  relation 

112 


x—a     x—b     x—y' 

\j  7.  Prove  that  the  middle  point  of  an  involution  is  always 
real. 

V  8.  What  is  the  form  of  an  involutoric  transformation  if  the 
double-points  are  +a  and  —a?  ^  'f   ~       "^^ 

"  9.  An  involutoric  transformation  referred  to  its  center  as 
an  origin  may  be  represented  by  x-y  =  k^,  where  ± k  locates  the 
double-points,  {k  may  be  real  or  imaginary.)  Prove  that  the 
anharmonic  ratio  of  the  points  represented  by  x,  y,-{-k,—  k  is  —  i. 

V  10.  Prove  that  the  rectangular  pair  of  an  involution  of  rays 
bisect  the  angles  formed  by  the  double-rays. 

II.  The  equation  of  a  circle 

is  given,  and  on  it  the  points  {+r,  o)  and  {—r,  o).  Find  the 
equivalent  equation 

vr—  kus  =  o, 

where  the  pencils  u—Xv  =  o  and  r—Xks  =  o  have  the  given  points 
as  vertices. 


§  8.   The  Complete  Quadrilateral.^ 

In  §  5  it  has  been  found  that  if  p  and  q  are  hnear  expressions 
in  X  and  y, 

^  See  Elemente  der  analytischen  Geometric  by  F.  Joachimsthal,  pp.  131-142. 


PENCILS  AND    RANGES.  27 

p  =  o, 
?  =  o, 

ap—^q  =  oy    \a 

) 

always  are  the  equations  of  four  harmonic  rays  of  a  pencil.     By 

means  of  this  theorem  it  is  now  possible  to  study  the  harmonic 

properties  of  the  complete  quadrilateral.     Let  ^=0,  q  =  o  and 


Fig.  9. 

/=o,  5  =  0  be  the  equations  of  two  pairs  of  rays,  Fig.  9.  The 
equations  of  the  rays  passing  through  the  vertices  of  these  pairs 
are  of  the  form  ap-\-^q  =  o  and  a'r-\-^'s  =  o  respectively.  For 
certain  values  of  ct,  /?  and  a',  /?'  these  equations  may  both  rep- 
resent a  ray  passing  through  the  vertices  of  both  pairs,  so  that 
we  have  the  identity 

(i)  ap+^q=a'r-\-^'s. 

From  this  the  further  identities 

(2)  ap—^'s  =  a'r—^q, 

(3)  ^'s—^q=ap—a'r 

follow.  Identity  (2)  represents  a  straight  hne  through  the  points 
of  intersection  of  the  rays  p=^o  and  5  =  0  and  of  the  rays  r  =  o  and 
q^o.  The  second  represents  a  line  passing  through  the  points 
of  intersection  oi  p  =  o  and  r  =  o  and  oi  s  =  o  and  ^  =  0.     Adding 


28  PROJECTIVE   GEOMETRY. 

(2)  and  (3)  we  get 

(4)  ap-(3q^ap-l^q, 

i.e.,  the  equation  of  a  ray  passing  through  O  and  P.  The  form 
of  the  equation  shows  that  the  ray  is  harmonic  to  the  ray  PQ 
with  regard  to  the  rays  PC  and  PD. 

Identity  (3)  may  be  written  ap—a'r==^'s—^q.  Subtracting 
this  from  (2),  there  results  the  new  identity 

(5)  a'r—^'s=a'r—^'s. 

This  is  the  equation  of  a  ray  through  O  and  Q.  The  form  of 
the  equation  shows  that  the  ray  QO  is  harmonic  to  the  ray  QP 
with  respect  to  the  rays  QB  and  QC.  As  (4)  and  (5)  result  from 
(2)  and  (3)  by  addition  and  subtraction,  it  is  proved  that  OP,  OQ 
and  AC,  BD  are  harmonic  pairs.  PC,  PD;  QB,  QC,  BD,  AC 
are  called  the  sides,  and  OP,  OQ,  QP  the  diagonals,  of  the  com- 
plete quadrilateral.  The  previous  results  may  be  summed  up  in 
the  theorem: 

In  every  complete  quadrilateral  a  pair  of  sides  always  forms  a 
harmonic  pencil  with  the  two  concurrent  diagonals. 

From  this  it  follows  that  two  vertices,  for  instance  C  and  D, 
are  harmonically  divided  by  the  two  diagonals  PO  and  PQ. 

Ex.  I.  If  p  =  o,  q  =  o,  r  =  o  are  the  equations  of  the  sides  of  a 
triangle,  prove  that  any  line  of  its  plane  may  be  represented  by 
an  equation  of  the  form  ap+^q-\-yr  =  o. 

Ex.  2.  Let  p  =  o,  q  =  o,  r  =  o  be  the  equations  of  the  diagonals 
of  the  quadrilateral,  prove  that 

ap+(]q+r^  =  o, 

~ap+^q+rr  =  o, 

ap—^q-\-jr  =  o, 

ap-\-^q—jr  =  o 

are  the  equations  of  a  quadrilateral  having  those  diagonals. 


PENCILS  AND   RANGES.  29 

§  9.   Perspective  Pencils  and  Ranges. 

In  §  6  it  has  been  found  that  the  equations  of  two  projective 
pencils  of  rays  with  the  vertices  5  and  T  may  be  written  in  the 
form 

^  U-\-  fiV=0, 

where  w  and  v  are  two  rays  through  S,  and  r  and  s  two  rays 
through  T.  In  general  the  product  of  these  pencils  is  a  curve  of 
the  second  order  with  the  equation 

(2)  us—rv  =  o. 

Every  value  of  p.  gives  two  corresponding  rays  u-\- nv  =  o  and 
r+iis  =  o,  which  intersect  each  other  in  a  certain  point  of  the 
curve.  •  We  will  now  assume  that  the  rays  w,  v  through  S  and  r,  5 
through  T  are  chosen  in  such  a  manner  that  there  exists  a  value 
k  oi  II  ?>o  that  the  two  corresponding  rays  u+kv  =  o,  r+ks  =  o 
are  identical,  or  that  the  ray  through  the  vertices  S  and  T  is  self- 
corresponding.     In  this  case 

(3)  it-{-kv  =  r+ks. 
Ehminating  ti  between  (3)  and  (2)  gives 

rs-\-ks^—ksv—rv=o,        or 

(4)  (r+ks){s-v)=o. 
Ehminating  v  between  (3)  and  (2)  gives 

^    2  ,  ^ 
us—rs—j-r^-\-Tur  =  o,        or 


(5)  (s+jrj{u-r)=o. 


Here  (x  =  —  X  and  s  =  ks  in  formulas  (3)  and  (4),  §  6. 


30  PROJECTIVE  GEOMETRY. 

Equations  (3),  (4),  and  (5)  show  that  in  this  case  the  curve 
of  the  second  order  degenerates  into  two  straight  Hnes,  one  pass- 
ing through  5  and  T,  the  other  passing  through  the  points  of 
intersection  of  w  =  o  and  r  =  o  and  of  1^  =  0  and  5  =  0.  Hence  the 
theorem : 

//  the  ray  connecting  the  vertices  of  two  projective  pencils  is 
self-corresponding,  then  the  product  of  the  two  pencils  consists  of 
the  self -corresponding  ray  and  another  straight  line. 

Two  pencils  of  this  kind  are  said  to  he  in  a  perspective  position, 
or  simply  in  perspective. 

Similar  arguments  in  line-coordinates,  which  may  be  left  as 
an  exercise  to  the  reader,  lead  without  difficulty  to  the  theorem: 

//  the  point  of  intersection  of  two  projective  ranges  is  self- 
corresponding  in  both  ranges,  then  the  product  {envelope)  of  these 
ranges  consists  of  the  self-corresponding  point  and  another  point. 

Two  ranges  of  this  kind  are  said  to  he  in  a  perspective  position, 
or  simply  in  perspective. 

The  line  where  corresponding  rays  of  two  perspective  pen- 
cils meet  is  called  axis  of  perspective.  The  point  through  which 
rays  joining  corresponding  points  of  two  projective  ranges  pass 
is  called  center  of  perspective. 

Ex.  Prove  the  proposition  concerning  perspective  ranges 
of  points  analytically  (line-coordinates)  and  geometrically. 

§  10.    General  Construction   of  Projective  Pencils  and  Ranges. 

In  §  2,  4  it  has  been  proved  that  a  projective  transformation 
is  determined  by  three  corresponding  pairs.  This  applies  to 
pencils  as  well  as  ranges.  This  fact  and  the  results  of  the  pre- 
vious section  make  it  possible  to  construct  projective  pencils 
and  rays. 

A.  Projective  Pencils. — Let  a,  h,  c  and  a' ,  b',  c'  be  three 
pairs  of  corresponding  rays  through  the  vertices  L  and  L'  re- 
spectively, Fig.  10.  These  determine  two  projective  pencils  of 
rays  through  the  points  L  and  L' .  Taking  c  and  c'  as  bearers 
of  two  ranges  of  points,  obtained  by  the  intersections  of  a' ,  h' , 


PENCILS  AND   RANGES. 


31 


c', . . .  and  a,  b,  c  with  c  and  </,  respectively,  we  have  accord- 
ingly the  projective  ranges 

(c-a'b'd  .  ..)=^{c'-ahc...). 

As  the  points  {cc')  and  {c'c)  are  identical,  it  follows  that  they 
are  in  perspective,  i.e.,  the  lines  joining  the  points  {ca')  and  {da)y 
{cb')  and  {c'b), .  .  .  are  all  concurrent,  say  at  P. 


Fig.   10. 


Hence,  if  any  ray  x  of  the  first  pencil  is  given,  we  know  that 
the  corresponding  ray  x/  will  be  situated  in  such  a  mariner  that 
the  Hne  joining  {xd)  and  {o^c)  will  pass  through  P,  and  x'  is 
found  by  joining  the  point  of  intersection  of  x  and  c'  to  P  by 
a  line  ^.    The  line  joining  U  to  the  point  of  intersection  of  f 


32  PROJECTIVE  GEOMETRY. 

and  c  is  the  required  ray  xf.  In  an  entirely  similar  manner 
any  ray  of  the  second  pencil  may  be  assumed  and  the  corre- 
sponding ray  in  the  second  pencil  be  constructed.  Any  ray  rj 
through  P  intersecting  c  and  c'  in  two  points  Y  and  Y'  gives 
rise  to  two  corresponding  rays  LY  and  VY' ,  or  y  and  y' .  From 
this  construction  it  is  seen  that  two  projective  pencils  always 
admit  of  a  third  pencil  which  is  in  perspective  with  each  of  them. 

Now  it  is  known  that  two  projective  pencils  produce  a  curve 
of  the  second  order  in  a  unique  manner.  The  six  rays  a,  b,  c] 
a',  h',  c'  determine  the  five  points  Z,,  L',  {aa'),  (66'),  {cc')  of  the 
curve,  and  every  new  pair  of  the  construction  like  :x;  and  x',  ;y 
and  y\  etc.,  determines  a  new  point  of  the  curve.  The  forego- 
ing construction  gives  us  therefore  a  means  to  construct  any 
number  of  points  of  a  curve  of  the  second  order,  as  soon  as  five 
of  its  points  are  given.  If  Z,  L' ,  A,  B,  C — in  any  order — are 
the  given  five  points,  join  L  and  U  each  to  ^,  J5,  C,  thus  ob- 
taining the  projective  rays  a,  h,  c  and  a',  ¥,  c' ;  then  apply 
the  construction  and  find  as  many  points  of  the  curve  as 
desired. 

The  ray  /  joining  L  and  L'  is  common  to  both  pencils,  but 
is  not  self -corresponding.  Suppose  t  belongs  to  the  pencil  at  L. 
To  find  its  corresponding  ray  at  V ,  produce  t  to  its  point  of  inter- 
section T'  with  c' \  join  T'  with  P  and  find  the  point  of  inter- 
section T  of  this  fine  with  c.  The  line  joining  1!  with  T  is  the 
required  ray  t' .  Following  this  construction  in  Fig.  lo  it  is  clear 
that  LIT  is  nothing  else  than  FL' .  Similarly,  if  t'  is  considered 
as  belonging  to  the  pencil  at  Z',  its  corresponding  ray  will  be 
FL.  Taking  a  ray  through  either  L  or  L\  very  close  to  ^,  and 
making  the  construction  for  the  corresponding  ray,  supposing 
at  the  same  time  that  the  original  ray  passes  to  the  Hmiting  posi- 
tion of  /,  it  is  easily  found  that  FL  and  F'L'  are  the  tangents 
from  F  to  the  curve  of  the  second  order. 

B.  Frojective  Ranges. — ^Let  A,  B,  C  and  A\  B' ,  C  be  three 
pairs  of  corresponding  points  on  the  lines  /  and  /'  respectively, 
Fig.  II.  These  determine  two  projective  ranges  of  points  on 
/  and  /'.     Taking  c  and  d  as  vertices  of  pencils  of  rays,  joining 


PENCILS  AND   RANGES. 


ii 


A',  B',  C, ,  and  A,  B,  C, . . . ,  respectively,  we  obtain  the 

projective  pencils 

{C-A'B'a...)  =  iC'-ABC...) 


Fig.  II. 


As  the  ray  CC  or  C'C  is  common  to  both,  it  follows  that  they 
are  in  perspective;  i.e.,  the  points  of  intersection  of  the  rays  CA' 
and  C'A,  CB'  and  C'B,  etc.,  are  all  on  the  same  straight  line, 
say  p. 

Hence,  if  any  point  X  of  the  first  range  is  given,  the  corre- 
sponding point  X'  is  found  by  joining  C  to  X  and  finding  the  point 
of  intersection  of  this  joining-line  with  p.  The  line  joining  C 
to  this  latter  point  cuts  V  in  the  required  point  X'.     In  an 


34  PROJECTIVE   GEOMETRY. 

entirely  similar  manner  any  point  of  the  second  range  may  be' 
assumed  and  the  corresponding  point  in  the  first  be  constructed. 
Any  point  in  p  gives  rise  to  two  corresponding  points  on  /  and  /'. 
From  this  construction  it  is  seen  that  two  projective  ranges  always 
admit  of  a  third  range  which  is  in  perspective  ivith  each  of  them. 
The  line  p  intersects  /  and  I'  each  in  a  point  whose  corre- 
sponding points  coincide  with  the  point  of  intersection  of  /  and  l\ 
Again,  /,  /',  AA',  BB',  and  CC  are  five  tangents  to  a  curve 
of  the  second  class  and  the  foregoing  construction  makes  it  pos- 
sible— by  joining  X  and  X' — to  construct  any  number  of  tan- 
gents. The  line  of  perspective  cuts  /  and  /'  in  their  points  of 
tangency. 

§  II.   Exercises  and  Problems. 

1.  Given  five  points  of  a  curve  of  the  second  order;  construct 
five  other  points,  each  being  situated  between  two  of  the  given 
points,  i.e.,  one  between  A  and  B,  one  between  B  and  C,  etc. 

2.  Construct  the  tangents  at  each  of  the  given  points. 

3.  Given  five  tangents  of  a  curve  of  the  second  class;  con- 
struct any  number  of  other  tangents  and  the  points  of  tangency 
of  the  given  tangents. 

4.  Two  projective  ranges  (ABC  ...)=- (A' B'C  ..  .)  on  the 
lines  I  and  /'  determine  a  curve  K  of  the  second  class  having 
AA',  BB',  CC,  ...  as  tangents.  Conversely,  every  tangent 
X  oi  K  cuts  /  and  /'  in  two  corresponding  points  of  the  ranges. 
If  we  now  turn  /'  about  its  point  of  intersection  P  with  I  through 
the  space  containing  K,  Fig.  11,  two  coincident  projective  ranges 
arise.  To  obtain  the  double  points  of  these,  §  2,  draw  the 
bisector  q  of  the  angle  between  I  and  /^  The  two  tangents  d^ 
and  c^2)  perpendicular  to  q,  intersect  either  /  or  V  in  the  required 
double-  or  self-corresponding  points  D^  and  Z>2-^ 

^  This  construction  has  been  successfully  used  as  a  base  for  the  synthetic 
treatment  of  the  projective  continuous  groups  by  Professor  Newson  and  myself. 
See  Kansas  University  Quarterly,  Vol.  IV,  p.  243  and  Vol.  V,  No.  i. 


PENCILS  AND   RANGES.  35 

5.  WhsLt  position  must  K  have  with  respect  to  I  and  I'  in  order 
to  make  the  projective  ranges  on  I  and  I'  involutoric? 

6.  Show  that  w^ith  K  as  a  circle  the  projective  ranges'  are 
involutoric. 

7.  Assume  five  points  L,  U,  A,  B,C  oi  a  curve  of  the  second 
order  in  such  a  manner  that  the  respective  pencils  are  involutoric. 

8.  \^erify  problems  i  and  2  on  a  given  circle. 

9.  Prove  Newton's  theorem  (Principia,  lib.  i.,  lemma  xxi). 
If  two  angles  AOS  and  AO'S  of  given  magnitude  turn  about 

their  respective  vertices  O  and  O'  in  such  a  way  that  the  point 
of  intersection  5  of  one  pair  of  arms  lies  always  on  a  fixed  straight 
line  u,  the  point  of  intersection  of  the  other  pair  of  arms  will 
describe  a  conic   (Cremona's  statement). 

ID.  Prove  AIaclaurin's  theorem  (Phil.  Trans,  of  the  Royal 
Society  of  London  for  1735). 

If  a  triangle  C'PQ  move  in  such  a  way  that  its  sides  PQ, 
QC,  C'P  turn  round  three  fixed  points  R,  A,  B,  respectively, 
while  two  of  its  vertices  P,  Q  slide  along  two  fixed  straight  lines 
CB',  CA',  respectively,  then  the  remaining  vertex  C  will  describe 
a  conic  which  passes  through  the  following  five  points,  viz., 
the  two  given  points  A  and  B,  the  point  of  intersection  B^  of 
the  straight  lines  AR  and  CB',  and  the  point  of  intersection  A' 
of  the  straight  lines  BR  and  CA\ 

§  12.    Projective  Properties  of  the  Circle. 

To  speciaHze  the  results  concerning  projective  pencils  for 
the  circle  it  is  simplest  to  depart  from  the  equation  of  the  circle 

(i)  x^+y^—r^  =  o. 

This  may  be  written  in  the  form 

(x-\-  iy)  (x—  iy)  —  r^  =  o, 

which  itself  may  be  considered  as  the  result  of  the  elimination 
of  A  between  the  projective  pencils 

{x+iy^Xr  =  o, 
(2)  \ 

[r+X{x—iy)=o. 


36 


PROJECTIVE   GEOMETRY. 


The  vertices  of  these  imaginary  pencils  are  the  points  of  inter- 
section of  the  Hne  at  infinity,  r  =  o,  with  the  rays  x-\-iy  =  o  and 
x—iy  =  o.  These  points  are  called  the  circular  points  at  infinity ^ 
Taking  the  center  of  the  circle  at  (a,  b),  the  equation  of  the 
circle  becomes 

(x—ay+{y—b)^—r^  =  o,     or 

(3)  {{x-a)  +  i(y-b)]  \(x-a)-i(y-b)]-r'  =  o. 

Eq.  (3)  is  the  result  of  the  elimination  between  the  projective 
pencils 

x+  iy—  (a+  ib)  +  Xr  =  o, 
r+  X{x—  iy—  a  +  ib)  =0, 


(4) 


and  shows  that  all  circles  of  the  plane  pass  through  the  same 
circular  points.  As  a  curve  of  the  second  order  is  determined 
by  five  points,  a  circle  must  be  determined  by  three  points,  two 
fixed  points  (the  circular  points)  being  given  in  advance. 

A  circle  can  also  be  produced  by  two  projective  pencils  with 
real   vertices.     Graphically   this   proposition   is   evident.     If,    in 

Fig.  12,  ST  be  a  chord  of  a  circle, 
all  angles  subtended  by  this  chord 
are  equal,  i.e.,  AASC=  lATC, 
ZBSC=  ZBTC,  etc.     Hence 

{abed  ...)  =  {a'b'dd'  .  .  .) ; 

the  pencils  at  S  and  T  are  projective. 
Connecting  any  point  in  space  with 
all  points  of  the  circle,  a  cone  is 
obtained.  Cutting  this  cone  by  any 
plane  and  passing  planes  through  the  vertex  of  this  cone  and 
the  rays  of  the  pencils  through  S  and  T,  two  new  pencils  of 
rays  {a.}>.^c^d^ .  .  .)  and  {a/b/c/d/  .  .  .)  are  obtained  on  the  inter- 
secting  plane,    which    are    again   projective.     Their   product   is 


Fig.  12. 


^  Introduced  by  Poncelet,  loc.  cit.,  p.  94. 


PENCILS  AND    RANGES.  37 

therefore  a  curve  0}  the  second  order;  in  this  case  a  conic.  It  will 
be  seen  later  on  that  all  curves  of  the  second  order  are  identical 
with  all  conies. 

To  show  how  a  circle  may  be  described  as  an  envelope,  assume 
first  the  Hne-equation  of  a  circle 

(5)  ^^+^'^=^, 


where  u  and  v  are  the  line-coordinates  and  r  the  radius.     Equa- 
tion (5)  is  the  product  of  the  two  projective  ranges 


(6) 


u+iv-{--  =  o, 
r 

—  +  A  (ic—iv)  =0. 
r  ^ 


The  coordinates  of  the  line  at  infinity  are  u  =  v=o\  conse- 
quently the  points  u^iv  =  o  and  u—iv  =  o  are  situated  on  the  line 

at  infinity.  —  =  0  is  the  equation  of  the  origin.  Thus  the  pro- 
jective ranges  (6)  are  situated  on  the  two  imaginary  straight  lines 
joining  the  points  u-\-iv  =  o  and  u—iv  =  o  with  the  origin.  These 
lines  are  tangent  to  the  circle  and  they  pass  through  the  circular 
points  at  infinity.  A  translation  does  not  change  these  results, 
so  that  the  theorem  may  be  stated : 

The  tangents  to  a  circle  from  its  center  pass  through  the  circular 
points  at  infinity.  , 

In  the  case  of  real  projective  ranges  producing  a  circle  it  is 
more  convenient  to  assume  the  circle  and  to  prove  that  it  is  the 
product  of  two  projective  ranges.  Let  in  Fig.  13  /LOAA'  =  a^ 
Z.OBB'=^,  ZAOB=-(f),  and  in  a  similar  manner  ZOA'A=a', 
AOB'B=^',  A  A' OB' =  4)'.  There  is  2a^2a'  =  7z-r,  2/?+ 2/5'  = 
Ti—y,  hence  /5— a=a'— /?'.  But  ^—a  =  (f)  and  a'— /5'  =  0',  hence 
<t>  =  4>'-  This  is  true  for  any  two  tangents  to  the  circle,  so  that  the 
pencils  (0-ABCD...)  and  {0-A'B'C'D'...)  are  projective. 
From  this  follows  that  the  ranges  formed  by  the  points  of  inter- 


38 


PROJECTIVE   GEOMETRY. 


section  of  all  tangents  with  two  fixed  tangents  are  equal.  Con- 
versely, the  product  of  these  particular  ranges  is  a  circle,  as  might 
also  be  proved  directly.  The  tangents  and  ranges  of  this  exam- 
ple may  again  be  connected  to  a  point  in  space.  Cutting  this 
configuration  by  any  plane  in  space,  two  projective  ranges  pro- 


FiG.  13. 


ducing  a  curve  of  the  second  class  are  obtained.  It  will  be  seen 
later  on  that  all  curves  of  the  second  class  are  identical  with  all 
conies  or  curves  of  the  second  order. 


§  13.   Polar  Involution  of  the  Circle. 

Through  a  given  point  A,  Fig.  14,  draw  any  ray  intersecting 
a  given  circle  in  two  points  C  and  D.  On  this  ray  determine  a 
point  B  in  such  a  manner  that  the  anharmonic  ratio 

(ABCD)  =  -i, 

i.e.,  harmonic.  If  this  operation  is  repeated  for  every  ray  pass- 
ing through  A,  the  points  B  on  all  these  rays  will  form  a  certain 
locus  which  is  a  straight  Hne,  and  which  is  called   the  polar  of 


POLAR  INVOLUTION. 


39 


the  point  A  with  regard  to  the  given  circle.     The  point  A  is  called 
the  pole.     To  prove  this  assume 
Y 


(I) 

as  the  equation  of  the  circle  and  (a,  o)  as  the  coordinates  of  the 
point  A .  The  special  position  of  point  and  circle  has  no  influence 
upon  the  generality  of  the  result.  The  equation  of  any  ray 
through  A  may  be  written 


(2) 


y=(a—x)m. 


Solving  (i)  and  (2)  it  is  found  that  the  abscissa  x^  and  X2  of  the 
points  of  intersection  C  and  D  of  the  ray  with  the  circle  are 


(3) 


Now 


or 


r-\-  am"^ + ^/  r^  +  2arm^  - 

-a^m^ 

I  +  W^ 

r-\-am^—\^r^-{-  2arm^- 

-a^m^ 

(AB'X.X,)  --=  (ABCD)  =  - 1, 


b—x^ 


b—Xo 


from  which 


h  = 


(J/  I  Jv-i  ~\      "^o  /  J^Jv-t-Jvn 

2a—{x^+x.^) 


4°  PROJECTIVE   GEOMETRY. 

Substituting  the  values  for  x^  and  X2  in  this  expression,  then 

ar 

b= ; 

a—r 

i.e.,  the  abscissa  of  B  is  independent  of  m  and  is  therefore  a  con- 
stant. The  locus  of  the  point  B  is  consequently  a  straight  line 
parallel  to  the  ;y-axis,  or  perpendicular  to  the  line  joining  the  point 
A  with  the  center  of  the  circle. 

If  A  is  without  the  circle,  there  are  rays  which  do  not  cut  the 
circle,  or  for  which  the  points  of  intersection  are  imaginary.  This 
is  the   case    when  r^+2arm^—a^m^<o,   or  a^ni^—2arm^>r'^,   or 

r  ... 

x^   and   X2   are   conjugate-imaginary,    so   that 


\m\  > 


"s/a^—  2ar 

X1+X2  and  x^x^  are  real  quantities  and  consequently  also  &  is  a 

real  quantity.     Hence  if  C  and  D  are  imaginary  B  is  still  real, 

±r 
and  (ABCD)  =  —  i.     If  m  =     . —  ,  the  points  C  and  D  coin- 

\/a^—2ar 

cide  and  the  rays  through  A  become  tangent  to  the  circle,  which 

are  real  when  A   is  outside  (2r<  a),  and  imaginary  when  A   is 

inside  (2r>a).     Hence  the  theorem: 

The  polar  of  a  point  with  regard  to  a  circle  passes  through  the 
points  of  tangency  [real  or  imaginary)  from  this  point  to  the  circle. 

In  the  case  of  a  pole  within  the  circle  the  equation  of  the 
polar  becomes 

a 

—  I 
r 

The  greatest  value  for  a  is  in  this  case  2r,  so  that  up  to  this 

limit i<i  and  h>a.     The  smallest  value  of  ^  is  for  fl  =  2r, 

r 

i.e.,   b  =  2r.      For  a<2r,  we  have   therefore    always    b>2r;   the 

polar  does  not  intersect  the  circle.     For  a  =  2r,  b  =  2r,  the  pole 

coincides  with  the  polar,  which  in  this  case  becomes  a  tangent; 

i.e.,  a  tangent  is  the  polar  of  its  point  of  tangency  and  a  point  of 

tangency  is  the  pole  of  the  corresponding  tangent.     For  the  center 

of  the  circle  a  =  r  and  &=  00 ,  the  polar  is  the  Hne  at  infinity.     For 


POLAR   INVOLUTION.  41 

the  tangents  from  the  center  m=  ±i(a  =  r),  so  that  the  equations 
according  to  (2)  become  x-\-iy  =  r,  x—iy  =  r.  This  shows  again 
that  the  tangents  from  the  center  oj  a  circle  touch  the  circle  at  its 
circular  points,  a  result  obtained  in  the  previous  section. 

§  14.    Continuation  of  §  13. 

Taking  any  point,  for  instance  B,  on  the  polar  of  yl,  it  is 
clear  that  the  polar  of  B  must  pass  through  A,  since  A  is  har- 
monic to  B  with  regard  to  C  and  D  as  the  other  pair.  Thus 
the  theorem: 

The  polar  oj  a  point  which  is  situated  on  the  polar  oj  another 
point  passes  through  the  latter  point.  Conversely,  the  pole  oj 
a  straight  line  which  passes  through  the  pole  oj  a  second  line  is 
situated  on  the  latter. 

From  this  it  follows  that  the  tangents  at  C  and  D  intersect 
each  other  in  a  point  of  the  polar  of  A .     This  point  is  evidently 


Fig.  15. 

the  pole  of  the  ray  (A BCD),  through  A.  Using  the  results  of 
§  8,  concerning  the  complete  quadrilateral,  it  is  now  easy  to  give 
a  simple  construction  of  the  polar  of  a  point,  or  of  the  pole  of  a 
straight  line.     Through  A   draw  any  two  rays  intersecting  the 


42 


PROJECTIVE   GEOMETRY. 


circle  in  the  points  C,  D  and  E,  F,  Fig.  15.  Connect  C  with  F, 
and  D  with  E,  and  find  the  point  of  intersection  G  of  these  con- 
necting Hnes.  In  the  same  manner  find  the  point  of  intersection 
H  of  the  lines  connecting  C  with  E,  and  D  with  F.  The  line 
through  G  and  H  is  the  required  polar  of  A .  The  proof  is  imme- 
diate, for  (ABCD)  =  (AEIF)  =  —  i,  which  is  the  condition  that 
GH  be  the  polar  of  A .  The  polar  of  H  must  pass  through  A , 
and  since  (HGBI)  =  —  i,  it  follows  that  it  also  passes  through  G. 
Hence  the  polar  of  HisA  G.  The  polar  of  G  passes  through  A 
and  H,  hence  AH  is  the  polar  of  G.  The  triangle  AGH 
possesses  the  important  property  that  the  polar  of  each  of  Us 


Fig.  16. 


vertices  is  the  opposite  side  in  the  triangle,  and  the  pole  of  each 
side  is  the  opposite  vertex  of  this  side.  This  triangle  is  called 
a  self-polar  triangle  with  regard  to  the  circle. 

Consider  now  in  Fig.  16  the  pole  P  and  its  polar  p  inter- 
secting the  circle  in  two  points  A  and  B.  Through  P  draw 
any  ray  c  intersecting  p  in  C,  and  determine  the  pole  C  of  the 


POLAR  INVOLUTION. 


43 


ray  c.  Then  {ABCC')  =  —  i.  Designating  the  tangents  from 
P  to  the  circle  by  a  and  h,  and  the  ray  PC,  which  is  the  polar 
of  C,  by  d ,  there  is  also  {ahcc')  =  —  i.  For  every  ray  through  P  a 
pair  of  poles  and  a  pair  of  polars  are  obtained  which  are  harmonic 
to  A  and  B,  and  to  a  and  h,  respectively.  In  this  manner  an  involu- 
tion of  coincident  poles  and  polars  arises.  In  the  case  of  the  figure 
A  and  B  are  the  real  double-points,  a  and  b  the  real  double- 
rays  of  the  involution.  It  is  noticed  that  in  this  hyperbolic  invo- 
lution each  pair  is  separated  by  the  double-elements.  Two  pairs 
either  exclude  each  other  entirely,  like  CC  and  DD',  or  include 
each  other  entirely,  Hke  DD'  and  EE'.  If  P  were  within  the 
circle,  we  should  have  an  elliptic  involution,  where  two  pairs 
always  overlap  each  other.  As  an  interesting  example  of  this 
kind,  consider  the  right-angle  involution  of  the  circle,  Fig.  17. 


Fig.  17. 

The  polar  of  the  center  is  the  Hne  at  infinity.  To  every  diam- 
eter a  as  a  polar  corresponds  a  pole  A  which  is  the  infinite  point 
of  the  perpendicular  diameter  a\  Thus  a  and  a'  are  a  pair  of 
the  polar  involution  about  the  center.  In  fact  the  rays  of  each 
pair  are  perpendicular  to  each  other.     To  find  the  double-rays  let 

y=mx, 


y= Xy 


44  PROJECTIVE   GEOMETRY. 

be  the  equations  of  any  pair.     For  a  double-ray  these  equations 

must  be  identical.     This  is  only  possible  when  w  = ,  or  ni^ 

=  — I,  which  gives  as  the  only  possibilities  m^  =  i,  m2=  —  i.  The 
equations  of  the  double -rays  are  therefore  x^iy  =  o  and  x—iy  =  o. 
As  they  are  the  double -rays  of  a  right- angle  involution,  the  para- 
doxical result  is  obtained  that  each  of  these  rays  is  perpendicular 
to  itself.     Geometrically  this  has  no  meaning. 

Ex.  I.  Construct  a  self -polar  triangle  having  two  poles 
within  the  circle. 

Ex.  2.  Discuss  the  elliptic  pole  and  polar  involution  and 
make  the  necessary  constructions. 

Ex.  3.  Explain  the  involutoric  relation  between  an  inscribed 
quadrilateral  A  BCD  of  a  circle  and  the  quadrilateral  circum- 
scribed Sit  A,  B,  C,  D. 


CHAPTER  II. 

COLLINEATION. 

§  15.    Central  Projection.^ 

A  central  projection,  or  a  perspective,  is  determined  by  the 
plane  of  projection  (plane  of  the  picture)  and  the  center  (eye). 
Assuming  the  plane  of  the  paper  as  the  plane  of  projection  and 
any  point  in  space  as  the  center,  it  is  possible  to  construct  the 
perspective  of  any  figure  in  space  on  this  plane. 

The  center  can  most  easily  be  located  by  a  circle  in  the 
plane  of  projection.  The  radius  of  this  circle  is  the  distance 
of  the  center  from  the  plane,  and  the  center  of  the  circle  is  the 
orthographic  projection  of  the  center  upon  the  plane  of  pro- 
jection. This  circle  has  been  introduced  into  geometry  by  Pro- 
fessor Fiedler  of  Zurich,  who  calls  it  distance-circle  ^  (Distanz- 
kreis).  In  this  section  only  the  projections  of  figures  in  a  plane 
will   be  considered    and  the  geometrical  laws    involved  in  this 

*  Historic  Note. — Desargues,  whom  Poncelet  called  the  MoNGE  of  his  cen- 
tury, was  the  first  to  investigate  the  relation  of  central  projection  to  the  geometry  of 
position;  i.e.,  the  purely  projective  properties  of  central  projection  (perspective), 
in  his  Methods  universelle  de  mettre  en  perspective  les  ohjets  donnes  reellement 
(Paris,  1636).  These  principles  are  also  contained  in  the  (Euvres  de  Desargues 
reunies  et  analysees  par  Poudra,  Paris,  1864,  Vol.  I. 

Brook  Taylor's  New  Principles  of  Linear  Perspective,  London,  1715  and  1719, 
and  J.  H.  Lambert's  work,  Die  jreie  Perspektive,  oder  Anweisung  jeden  perspekti- 
vischen  Aufriss  von  freien  Stiicken  und  ohne  Grundriss  zh  verfertigen,  Zurich,  1759, 
II.  part,  1774,  contain  also  the  fundamental  principles  of  perspective. 

For  further  information  see  the  introductory  chapter  of  Wiener's  Darstellende 
Geometric,  Leipzig,  1884-87,  which  contains  a  history  of  this  science  and  a  chapter 
on  perspective  in  Vol.  II;  also  Fiedler's  Darstellende  Geometric,  Vol.  I. 

'  D.  Geometric,  Vol.  I,  1883,  and  Cyclographie,  Chapter  VIII.  The  method 
followed  here  is  that  of  Fiedler. 

45 


46  PROJECTIVE   GEOMETRY. 

projection  explained.  The  plane  of  projection  will  be  desig- 
nated by  tt',  and  the  arbitrary  plane,  whose  perspective  will 
be  made,  by  tz,  Fig.  i8.  Let  s  be  the  hne  of  intersection  of  n 
and  Ti'.  To  obtain  the  projection  P'  of  any  point  P  in  tt,  con- 
nect P  with  the  center  C  and  determine  the  point  of  intersec- 
tion P'  of  this  connecting  line  with  u' .  In  a  similar  manner, 
the  projection  /'  of  a  line  I  (RS)  in  tt  is  obtained  as  the  hne  of 
intersection  of  the  plane,  passing  through  C  and  /,  with  t:'. 
From  this  construction  the  following  fundamental  laws  are 
immediately  clear: 

To  every  point  of  tz  corresponds  a  point  of  tz' ,  and  conversely, 
and  both  points  lie  on  a  ray  through  C. 

To  every  straight  line  of  tz  corresponds  a  straight  line  of  tz',  and 
conversely,  and  both  lines  meet  in  a  point  of  s. 

To  the  line  at  infinity  of  tz  corresponds  a  line  q'  of  tz'  which  is 
parallel  to  s.  Conversely,  to  the  line  r'  at  infinity  of  tz'  corresponds 
a  line  r  parallel  to  s. 

The  plane  tz  is  usually  determined  by  its  trace  5  in  tz'  and 
either  of  the  lines  r  and  q' .  If  a  straight  line  /  in  tt  is  given,  inter- 
secting 5  in  S,  the  corresponding  hne  I'  is  determined  by  drawing 
a  Hne  through  C  parallel  to  I  and  marking  its  point  of  intersection 
Q'  with  q'.  It  is  apparent  that  Q'  is  the  projection  of  the  infinite 
point  of  I,  and  the  projection  of  /  consequently  passes  through  S 
and  Q'.  Another  way  is  to  produce  /  to  its  point  of  intersection 
R  with  r  and  to  join  C  with  R.  The  line  through  S  parallel  to 
CR  is  I'.  From  the  figure  it  is  seen  that  CRSQ'  is  a  parallelogram 
and  that 

PS:PR  =  P'S:CR. 

The  planes  through  C  parallel  to  tz  and  tz'  form  a  space  of  a 
parallelepiped.  Keeping  tz'  fixed,  it  is  possible  to  turn  the  planes 
TZ  and  the  planes  through  C  parallel  to  tz  and  tz'  down  into  tz, 
without  changing  5  and  q'  and  the  distances  of  C  and  r,  C  and  q', 
S  and  r,  and  5  and  q'  in  these  planes. 

After  the  motion  there  is  still  CR  \\  and  =  Q'S,  and  SP'=SP'; 
consequently  the  distances  PR  and  PS  are  not  changed  by  the 


COLLIN  RATION. 


47 


motion.  From  this  it  follows  that  after  the  motion  P'  and  the 
revolved  position  of  P  he  on  a  ray  through  the  revolved  position 
of  C.  The  laws  expressing  the  geometrical  relation  between  the 
revolved  and  the  projected  figure  are  therefore  the  same  as  those 


Fig.  i8. 

between  the  figure  in  space  and  its  projection.     After  the  rabatte- 
ment,  Fig.  i8  assumes  the  form  of  Fig.  19. 

In  this  figure  I  and  /'  are  the  two  corresponding  lines  which 
with  5  and  SC  form  a  pencil  of  four  rays  through  S.  As  this 
pencil  is  intersected  by  the  rays  CP  and  CQ,  we  have 

{CLP'P)  =  {CMQ'Q). 


The  value  of  iCMQ'Q)  is 


CQ'      CO 


=  k,  say;    i.e.,  entirely  in- 


MQ'    NO 

dependent  of  the  position  of  /,  /',  and  CP.  Thus,  drawing  any 
ray  through  C,  intersecting  s  in  S,  and  constructing  any  two 
corresponding  points  P  and  P'  (rotated  position  of  a  point  in  tz 
and  its  projection  on  n'),  we  have 

{CSPP')=  const. 


48  PROJECTIVE   GEOMETRY. 

Keeping  CS  fixed  and  constructing  all  possible  pairs  (P,  P'), 
two  coincident  projective  series  of  points  are  obtained  having  C 
and  5  as  double-points.  The  different  cases  of  central  projection 
may  be  classified  according  to  the  position  of  the  center  and  to  the 
value  of  the  constant  k  of  the  projection.     Before  entering  upon 


Fig.  19. 

these  details  it  is  important  to  establish  the  analytical  relation 
between  a  pair  P,  P' . 

Ex.  I.  In  both  figures  18  and  19  it  is  noticed  that  the  distance 
between  ;'  and  s  is  equal  to  the  distance  between  C  and  q'.  As- 
sume the  elements  of  a  perspective  as  in  Fig.  19,  and  draw  the 
perspective  of  a  triangle  (a)  which  does  not  cut  r\  also  of  a  triangle 
(b)  which  cuts  r. 

Ex.  2.  Draw  in  a  similar  manner  the  perspective  of  a  regular 
hexagon  which  is  not  cut  by  r. 

Ex.  3.  Draw  the  perspective  of  a  circle 

{a)  which  does  not  cut  r\  also  of  a  circle 

(b)  which  cuts  r  in  two  points;  and  of  a  circle 

(c)  which  touches  r. 


COLLIN  RATION.  49 

Ex.  4.  Draw  the  perspective  of  a  system  of  concentric  circles. 

Ex.  5.  Find  I'  when  I  is  parallel  to  s. 

Ex.  6.  Construct  the  perspective  of  a  circle  having  its  center 
at  C,  Fig.  19. 

Note. — In  all  these  exercises  the  given  figures  are,  of  course, 
in  the  revolved  position  of  n]   i.e.,  in  ■k'. 


§  16.   Analytical  Representation  of  Central  Projection. 

In  Fig.  19  assume  any  two  perpendicular  lines  through  C  as 
coordinate  axes  and  designate  the  angle  which  the  X-axis  makes 
with  CO  by  ^,  and  its  angle  with  CP  by  <!>.  Designate  the  coor- 
dinates of  P  and  P'  respectively  by  x,  y  and  x',y'.    Now 

(CLPF)  =  k     or    CF  =  ^-^^     or    CF^^P^CP'-CL) 


CPCL 
From  this  CP' == 


k-CL-{k-i)CP' 
CN 


Now  CP=^/x'-+y\    CL  = 


7-7 — jr,  or,  smce 

cos  {(p—cpy 


,  ,      , .  ,  ,       .      ,    .     ,      X  cos  d)        o'  sin  0 

cos  {(p-  (j))  =cos  0  cos  <^+sm  ^  sm  (^  =  —===+  -/=^„, 

Vx^+y^    Vx^+y^ 

CNVx^+f 


CL  = 


X  cos  (I'+y  sin  ^' 
hence,  by  substitution  in  the  above  value  for  CP', 

CN-\/o^+f 


CF^ 


(i  —  k)  cos  (p-x+(i  —  k)  sin  (p-y+k-CN' 


50  PROJECTIVE   GEOMETRY. 

Now^^CP'-cos  ^;  /=CP'-sin(^;  hence 


(I) 


CN-x 

0(f  = 


y- 


{i  —  k)  cos  (l)-x-^{i  —  k)  sin  (p-y+k-CN' 

CN-y 

(i  — ^)  cos  (p-x+{i  —  k)  sin  (p-y+k-CN  ' 


In  these  expressions  there  are  three  arbitrary  parameters :  CN, 
k,  (f).     Conversely,  if  the  transformation 


(11) 


^  ax 

dxArey-\-f 
ay 
■^      dx+ey+j 


is  given,  it  alv^ays  represents  a  perspective.     To  prove  this,  it  is 
sufficient  to  reduce  (II)  to  the  form  (I).      This  can  be  done  in 

one  and  only  one  way,  by  putting  —  =  k, 

(i  —  k)  cos  (f)    d       (i  — ^)sin0    e 


e 
and  as  a  consequence  -7-  =  tan  (p. 


CN  a'  CN  a' 

e 
J 


From  this  CN=—.       — .     Equations  (II)  are  the  most  general 

Ve'^  +  d^ 

representation  of  a  perspective.  The  points  {x,  y)  in  n  for  which 
(x',  /)  in  tt'  become  infinite  are  evidently  situated  in  the  line 
dx+ey-\-}  =  o.  This  is  therefore  the  equation  of  the  line  r.  For 
the  hne  5  we  have  x=x/,  y=y'  \  hence  from  the  first  equation  of 

(11) 

dx"^  +  exy  +  /x  =  ax, 
or  dx^ey\]—a  =  o, 

as  the  equation  of  s. 

Equations  (II)  may  also  be  written  in  the  form 

x{dxf  —  a)  +  yex^  +}x'  =0, 
xdy'  +  y{ey'  —  a)  +  jy'  =  o. 


COLLINEATION.  51 

The  condition  that  the  values  for  x  and  y  become  infinite  is 


=0, 


{dxf—a)         ex/ 
dy         {ey'—a) 
or  explicitly 

doc'-\-ey'-{-a=o. 

This  is  therefore  the  equation  of  q'. 

f-a 
In  these  calculations  the  coordinate-origin  is  C,  so  that 


Vd'+e"" 

is  the  distance  of  5  from  C,  or  CN. 

a 
The  distance  of  a'  from  C  is     ,  ,  and  that  of  r  from  C  is 

Vd^+e^ 

, .     This  naturally  all  agrees  with  Fig.  19  from  which  they 

were  derived.  It  must  be  remarked  that  these  formulas  only 
hold  when  C  is  in  finite  regions. 

§  17.   Special  Cases  of  Central  Projection. 

A.  Involution. — ^If  in  Fig.  i8  the   center  C  is  situated  in 

the  bisecting  plane  of  n  and  n'  and  if  n  is  subsequently  turned 

in  the    direction  of  the  space   between  n   and  ii'  in  which   the 

bisector  Hes,  then  r  will  coincide  with  5'',  and  after  the  rotation 

CO  =  —  NO.     C  and  g',  r,  in  this  case,  are  on  the  same  side  of  s\ 

CO 
k=-:rjz:z  =  —  i.     This   perspective   is   called  an  involution,   since 

in  Fig.  19  to  every  line  /  of  ;z-  corresponds  a  line  I'  of  n'.  If  /'  is 
considered  as  the  rotated  position  of  a  line  in  tt,  then  its  corre- 
sponding Hne  in  7:'  when  rotated  will  coincide  with  /.  This  fact 
immediately  appears  from  the  construction  and  also  from  equa- 
tions (II),  which  in  this  case  assume  the  form 


(ni) 


ax 

X 


dx+ey—a* 

ay 
dx+ey—a* 


52 


PROJECTIVE  GEOMETRY. 


or 


dxo(f—a{x-\-  xf)  +  ex'y  =  o, 
eyy'  —«(>'+/)  +  dxy'  =  o. 


Now  in  every  perspective  o</y  =  xy',  as  is  seen  immediately  by 
dividing  both  equations  (II) ;  hence  these  equations  remain  the 
same  when  x,  y  and  a/,  y'  are  interchanged. 

If  the  plane  it'  is  turned  in  the  opposite  direction  as  in  the 
involution,  ^  =  + 1 ,  and  the  revolved  position  of  n'  in  this  case 
is  also  obtained  from  that  of  an  involution  by  a  reflection  on  the 
5-axis. 

B.  Similitude. — This  case  arises  when  n'  \Tt\  i.e.,  if  5  is 
infinitely  distant;  hence  the  equation  of  s  must  appear  in  the 
form  }—a=o  and  d=o,  e=o.     Equations  (II)  now  go  over  into 


(IV) 


y=jy. 


CP' 


When  ^  =  -  is  positive,    {CLP'P)  >  o   and    equals    ^:^  =  k. 

P  and  P'  are  on  the  same  side  of  C,  and  the  center  in  space  is  on 
the  same  side  of  tt'  and  n,  Fig.  20a. 


Fig.  20a. 


If  k  is  negative,  P  and  P'  are  on  different  sides  of  C,  and  the 
center  lies  between  tt'  and  tz.  li  k  =  —  i,  the  center  is  in  the 
middle  of  n^  and  tt  and  the  perspective  becomes  central  symmetry, 
Fig.  20b. 


COLLIN  RATION.  53 

C.  Affinity.^ — By  this  term  we  designate  a  perspective 
whose  center  is  at  an  infinite  distance.  All  rays  through  C  are 
parallel  and  intersect  the  axis  of  collineation  5  at  a  constant 
angle.  To  prove  this,  draw  through  every  projecting  ray,  inter- 
secting iz'  and  7t  in  two  points  P  and  P' ,  a  plane  parallel  to  some 
fixed  plane,  and  intersecting  tt'  and  tt  in  the  lines  P'Q  and  PQ; 
where  Q  is  a  point  in  s,  Fig.  21a.     For  all  planes  of  this  kind 


Fig.  2 Iff. 

P'Q 

the  triangles  PQP'  are  similar,  i.e.,  ^^=  const.;     furthermore, 

for  every  plane  the  lines  P'Q  and  PQ  include  constant  angles 
with  s.  Hence,  after  revolving  iz  down  into  rJ ,  Fig.  21&,  and 
connecting  again  P    with  P',    A PQL ^consi.,    Z P'Q L  =  const., 

remains  parallel  to  some  direction  cutting  5  at  a  constant  angle. 

r.       PQ  ■  sin  (PQL)      „^ 
Now  PL=    ^.    ,„;  Jf,   ^  =PQ- const.; 

sm  (PLQ)  ^ 

^  Introduced  by  MoBlxJS  in  his  Barycentrische    Calcul,  p.  150. 


54 


PROJECTIVE  GEOMETRY. 


similarly 


hence 


„,^     P'Q  sin  (P'QL)     „,^ 

P'T  =  —  -    ^-  =  P'O  ■  const  • 

^^         sin(FL(2)  ^  ' 


P^ 
PL 


=  const. 


FL 


The  same  result  might  be   found  from  Fig.  19,  where  ^^  =  ^. 

Formulas  (II),  however,  are  not  valid  in  this  case.  To  estabhsh 
the  analytical  relation  between  P',  (x',  y')  and  P,  (x,  )/)  assume 
now  5  as  the  x-axis  and  any  perpendicular  to  it  as  the  )'-axis. 


Fig.  2I&. 


Let  the  constant  slope  of  PP'  be  w;  then  the  equation  of  the 
ray  through  P  is,  when  ^  and  -q  designate  current  coordinates, 
ri  —  y  =  in{z  —  x),  and  the  distance  of  L  from  O,  Fig.  21&,  is  obtained 

Now  from  the  figure  X—x'  =  —  k{x—X)\  eliminat- 


as  i  =  - 


m 


ing  X,  there  is  found 
(V) 


X  =x— y, 


m 


y'  =  ky. 


These  are  the  equations  of  afhnity.     Conversely,  if  a  transforma- 
tion 

0!/  =x+ay, 

y  ^hy 


(VI) 


COLLIN  RATION. 


55 


is  given,  it  may  always  be  represented  in  the  form  (V),  by  putting 

'b  =  k  and  - —  =  — w.     A  characteristic  property  of    this  trans- 
a 

formation  is  that  closed  curves  are  transformed  into  closed  curves, 
so  that  the  areas  enclosed  by  the  two  have  the  constant  ratio  k 
(in  V).  To  prove  this  assume  any  triangle  ABC  and  the  corre- 
sponding triangle  A'B'C.     Designate  the  points  of  intersection 


Fig.  22. 


of  AB  and  A'B',  BC  and  B'C,  CA  and  CA'  by  C^,A,,  B„  respect- 
ively.    Now  in  Fig.  22 

(i)  aABC=  aAC,B,+  aBA,C,+  aCB.A,, 

(2)  aA'B'C'=aA'C,B,+  aB'A,C,+  aC'B.A,; 

but  as  the  corresponding  triangles  on  the  left  side  have  equal 
bases  and  altitudes  of  constant  ratio  k,  we  have 

AA'C,B,  =  k-  aAC^B,;     AB'A^C,  =  k-  a5^A; 
AC'B^A^  =  k-  aCB.A^. 


Substituting  these  values  in  (2)  and  dividing  (i)  by  (2),  there 


IS 


AA'B'C'  =  k-AABC. 


Q.E.D. 


As  these  triangles  may  be  infinitesimal,  it  follows  by  limiting 
summations  that  the  same  property  holds  for  any  corresponding 


156  PROJECTIVE   GEOMETRY. 

areas.  If  k  is  negative,  it  follows  that  the  area  of  A'B'C  is  also 
negative.     For^=  — i 

(VII)  J'^^^^m^' 

which  represents  as  a  special  case  of  affinity  oblique  axial  sym- 
metry. For  w  =  00  this  goes  over  into  orthogonal  axial  symmetry. 
{x'  =  x,  y'  =  —  y)' 

When  y^=  +  i  and  m^o  an  identical  transformation  is 
obtained.  But  the  case  is  also  possible  where  ^==  +  i  and  w  =  o; 
i.e.,  where  the  rays  PP'  or  AA'  are  parallel  to  s,  Fig.  23.     In 


i-k 

this  case  

m 

tions  are  now 


can  have  any  value,  say  —X,  so  that  the   equa- 


(VIII)  jx;=x+A,, 

{y'  =  y. 

The  effect  of  this  transformation  is  that  every  point  is  moved 
parallel  to  s,  and  the  amount  of  the  motion  is  proportional  to 
the  distance  of  the  point  from  s.     As  in  (VII),  equal  areas  are 


COLLINEA  TION.  5  / 

here    transformed    into    equal    areas.     This    transformation    is 
called  elation} 


§  i8.   Exercises  and  Problems. 

1.  Given  a  straight  line  with  the  equation 

fx+qy-\-r  =  o. 

Find  the  equation  of  the  perspective  of  this  line,  and  discuss  its 
position  with  respect  to  the  original  line,  q' ,  C,  and  s. 

2.  Let  x'y'  =  l{oc')  be  the  equation  of  a  curve  and  suppose 
that  j{oc')  is  not  divisible  by  x'  and  remains  finite  iox  x/  =o\  then 
for  this  value  of  x\  y'  becomes  infinite;  in  other  words,  the 
curve  approaches  the  ;y-axis  asymptotically. 

Applying  the  transformation  (II)  to  this  equation,  or  making 
a  perspective  of  this  curve,  its  equation  becomes 

a^xy{dx-\-  ey+  /)'*~^  =  (f)(x,  y), 

where  (f)(x,  y)  is  a  polynomial  of  x  and  y  of  the  wth  order,  pro- 
vided f{x)  is  an  integral  algebraic  function  of  x  of  the  nth. 
degree.    For  x'  =  o,  y  =  oo  .    The  corresponding  values  of  x  and  y 

are  x  =  o  and  y= .     Thus  putting  in  the  above  equation  x  =  o, 

the  value  of  y  is  found  to  be .     The  infinite  branch  of  the 

curve  is  therefore  transformed  into  a  finite  branch.  This  is  gen- 
erally true,  as  it  follows  directly  from  equations  (II) .     If  ic'  =  oo  ; 

y  =  oo,  --}  =  m  (finite),  the  corresponding  point  x,  y  is  necessarily 

situated  in  the  line  r,  whose  equation  is  dx-{-ey-\-f  =  o.      Now 

y         y' 

—  =  --j  =  m\   hence  y  =  mx  and  from    dx+emx-{-f=o,  the  coor- 

/  mf 

dinates  ^=  — tt — ,  y=~TZ —  of  the  required  point  are  found. 

^  Term  used  by  S.  Lie,  loc.  cit. 


58  PROJECTIVE  GEOMETRY. 

3.  Find  the  transformed  equation  of  xy=i  and  discuss  it. 

4.  Find  the  transformed  equation  of  the  circle 


^  +}'  = 


^2+e2 


5.  Transform  perspectively  the  curve  y'  =  e^.      For  5c/  =  oo 

9  ^  / 

%  OC  A/  T 

/  =  Go.     Now    3'=i  +  x+^  +  -j+  ...  ;    hence     -  =  - +  1  + 
—+  —  +.,.  and  lim  (  —  )  =  00  .     By  equations  (II), 


(^y ^    dx+ey+f 


dx+ey-\-f 
Now  for  x'  =  CO  ,     y'=cc  ,     we  have 


—  =  -7=00     and     — =0 

X    x'  y 


d  dA-    —  A-i-  ^ 

hence      y  =  ^      '^^      becomes     r=e°=i: 

y       y  y 

hence     y  =  and     x=o. 

Make  the  corresponding  construction. 

6.  A  perspective  does  not  change  the  degree  or  class  of  a  curve. 
As  the  curve  is  supposed  to  be  algebraic,  we  may  represent  it 
by  the  polynomial  }(x,  y)=o,  which  evidently  does  not  change 
its  degree  when  transformed.     Show  this  directly. 

■7.  Prove  analytically  that  the  transformation 

x'  =  x-{-ay, 
y'  =  hy 

transforms  the  area  of  the  triangle  ABC  {x^,  y^\  x^,  y2',  oc^,  ys) 
into  an  area  of  the  triangle  A'B'C  (x/,  y^ ;  x-l ,  y^ ;  x^ ,  y^), 
so  that  £\A'B'C' =  !)•  aABC.     Use  determinants. 


COLLIN  EATION.  59 


§  19.    Collineation.^ 

In  the  last  two  sections  it  has  been  seen  that  every  perspec- 
tive transformation  transforms  a  straight  Hne  into  another  straight 
line  (into  a  point  if  the  line  passes  through  the  center).  The 
question  is  now  whether  there  are  other  transformations  with  this 
property.  From  analytical  geometry  it  is  known  that  transla- 
tion and  rotation  are  transformations  of  this  kind. 

A.  Translation! — By  this  operation  all  points  of  a  plane 
are  moved  parallel  to  a  certain  fixed  direction  by  the  same  amount. 
The  equations  are 


0^)  \y-y+i. 

The  slope  of  the  direction  in  which  (x,  y)  is  moved  is 

y'—y    h 
xf—x    a' 


.and  the  amount  \^a~+b^. 

B.  Rotation. — If  every  point  (x,  y)  is  turned  through  an 
angle  (f)  about  the  center  (origin),  the  coordinates  oc',  y'  of  the 
rotated  point  may  be  expressed  by 

\x'  =  x  cos  <i>—y  sin  0, 
\y'  =  x  sin  0  +  >'  cos  <^. 

(IX)  and  (X)  are  the  equations  of  ordinary  motions  in  a 
plane.  If  to  a  point  (x,  y)  we  first  apply  a  rotation  (X)  and 
then  a  translation  x"  =  x'^a,  /'  =  /  +  &,  and  finally  writing 
again  o(f  and  y'  for  xf'  and  y"  ^  the  result  is 

jx'  =  xcos  <^—y  sin  0+a, 
(y  =  x  sin  <}!)+3' cos  ^+&. 

*  Called  by  Chasles,  in  his  Geometric  Suferieure,  art.  99,  homographie;  homo- 
graphic  means  to  be  in  collineation.  The  word  collineation  goes  back  to  Mobius' 
Barycentrische  Calcul. 


6o  PROJECTIVE  GEOMETRY. 

Such  a  transformation  changes  only  the  position  and  not 
the  shape  of  a  figure,  nor  its  size.  Another  transformation  which 
does  not  change  the  character  of  a  straight  hne  is 

C.  Dilatation. — ^The  equations  for  these  are: 

{oc'  =ax, 

and  may  physically  be  illustrated  by  stretching  a  piece  of  rubber 
first  in  the  direction  of  the  :r-axis  and  then  in  the  direction  of 
the  y-axis.  Equal  distances  along  one  of  the  axes  are  stretched 
by  the  same  amount.  If  one  of  the  coefhcients  a,  /?  is  i,  then 
(XI)  represents  an  orthogonal  affinity.  Combining  the  dilatation 
o</=ax,  y'=^y  with  the  affinity  :x/'=y  +  ai/, /'=/?i/  and  then 
dropping  one  prime  clear  through,  the  result  is 

oc'=ax+ai^y, 

Applying  to  this  the  rotation  x"  =  c^  cos  ^— /  sin  (f)+a', 
y"  =xf  sin  0  +  /  cos  (f)+i>',  and  dropping  one  prime,  the  result  is 

x'=q;  cos  cf)-x+  (ai/5  cos  0— /5/?i  sin  (f))y+a\ 
y'  =a  sin  (^-x-^ia^i^  sin  4>-^(^^i  cos  cf))y+y, 

representing  thus  a  combination  of  a  dilatation,  an  affinity,  a 
translation,  and  a  rotation.     Conversely,  every  transformation 

(  x:'  =ax-{-by-{-c, 
^^^^^  ly=dx+ey+j, 

represents  such  a  combination.     To  prove  this,  put  a=a  cos  (f), 

a  /- 

<^  =  2sin^;  i.e.,tan  ^  =  -7,  a=v  a^+c^^     Further, 

a^/?  cos  <f)—^8^i  sin  <f>  =  b, 
a^^  sin  0  +  /?,5i  cos  0  =  ^, 


COLLIN  EA  TION.  6 1 

from  which 

_     6cos^  +  esin<^     ,  ,         .     ,       bd+ae 

aM  = ,,,.■,■   =0  coS9+e  sin0=     .  , 

ed— ah 
pp-^  =  e  cos  cf)—b  sin  cj) 


Va^+d^ 

and  finally  a'  =  c,  &'  =  /. 

The  transformations  making  up  (XII)  all  leave  the  infinitely 
distant  line  unchanged  and  transform  areas  into  proportional  areas. 
The  same  is  therefore  also  true  of  their  combination.  Such  a 
transformation  is  called  a  Linear  Transformation,  or  Linear 
Deformation.  As  all  of  its  constituents  are  projective,  a  Unear 
transformation  is  also  projective,  i.e.,  it  transforms  pencils  and 
ranges  into  projective  pencils  and  ranges.  Perspective  and  linear 
transformations  are  two  of  the  most  important  projective  trans- 
formations. 

D.  General  Collineation. — A  perspective  contains  three 
arbitrary  parameters,  which  is  apparent  when  numerators  and 
denominators  of  equations  (II)  are  divided  by  a.  Applying  to 
the  point  x,  y  o,  linear  transformation 

x'  =  ax-{-by  +  c, 
y'=dx+ey+f, 

and  then  to  the  transformed  point  x',  y'  the  perspective 

^'^d,x!^e,y'^f^ 

l 

d.x'^-e^y  +  f^' 

the  result  is,  after  dropping  one  prime, 

ax-\-by-\-c 


y"  = 


(xiiio 


/= 


{d^a^  e^d)x\  (^ib  +  e^e)y+f^ 

dx+ey-\-f 
{d^a+  e^d)x+  {dj)+e,e)y+  f^' 


62  PROJECTIVE   GEOMETRY. 

These  equations  are  of  the  form 


(XIII) 


0(f  = 


Conversely,  every  expression  of  the  form  (XIII)  can  be  repre- 
sented by  (Xlir).  To  show  this,  put  a^  =  a,  b^  =  b,  c^  =  c\  d.^=dy 
h^  =  e,  C2=-/;   d^a^-e^d  =  a^,  d^^e^e  =  h^,  /i  =  «3-     From 

dji-^e.^d  =  a^^ 
d^h+e^e  =  \,   ■ 

a^e—h^d    a^h^-h^a^ 
we  una  »i  = "uj^ — x^ — rT» 

b^a—aj)     b^a^—agbi 
ae—  bd     a^b^—  b^a^' 


e,= 


By  means  of  these  formulas  it  is  possible  to  resolve  every  trans- 
formation (XIII)  into  a  perspective  and  into  a  linear  transforma- 
tion. The  principal  property  of  this  transformation  is  that  it 
transforms  every  straight  line  and  every  point  projectively  into  a 
straight  line  and  a  point.  It  is  the  general  projective  transfor- 
mation of  the  plane,  or  a  collineation  in  the  plane. 

Dividing  numerators  and  denominators  of  (XIII)  by  Cg,  it  is 
seen  that  a  colHneation  generally  depends  upon  eight  parameters. 
To  prove  that  this  is  the  most  general  transformation  which  trans- 
forms straight  lines  into  straight  hues,  assume  that 

^      'S(x,  y) 


COLLIN  EATION.  63 

be  a  transformation  of  this  kind;  then  the  equation  of  every 
straight  hne 

ax'+hy+c=o, 

where  a,  &,  c  may  have  any  real  values,  must  be  transformed  into 
a  linear  equation  between  x  and  y.     Thus 

aP{x,  y)  -Six,  y)  +  bR{x,  y)Q{x,  y)  +  cQ(x,  y)-S(x,  y)  =0 

must  be  linear  for  all  real  values  of  a,  b,  and  c.  This  can  only 
be  true  if  P,  Q,  R,  and  5  are  themselves  hnear  functions  of  x 
and  y. 

§  20.   Geometrical   Determination   and  Discussion   of 
Collineation. 

1.  The  equations  of  collineation  depend  upon  eight  parameters; 
these,  when  known,  determine  a  collineation.  If,  therefore,  any 
four  points,  of  which  no  three  He  in  a  straight  line,  are  given: 
'4i(:x:i,  >'i);  A^{x2,y2);  As(Xs,y3);  A,(x^,y^),  we  can  assume  any 
other  four  points  with  the  same  property  as  corresponding  points 
of  a  coUineation.  That  this  assumption  is  legitimate  is  seen 
from  the  eight  equations  which  may  be  established  between  the 
coordinates  of  the  given  points  A^,A2,  A^,  ^^and  ^/,  A2,  A^ ,  A/ 
by  formulas  (XIII).  The  eight  independent  parameters  are  now 
the  unknown  quantities  which  from  the  eight  equations  of  condi- 
tion may  be  extracted  in  a  definite  manner.     Hence  the  theorem: 

There  is  one  and  only  one  collineation  which  transforms  a  quad- 
rilateral in  a  plane  into  any  other  quadrilateral  of  the  same  plane. 
Two  quadrilaterals  in  a  plane  determine  a  collineation  uniquely. 

2.  An  important  problem  in  a  colhneation  is  the  determina- 
tion of  those  elements,  points,  or  straight  Unes  which  are  not 
changed  in  position,  i.e.,  of  the  invariant  elements.  To  find  the 
straight  lines  which  are  invariant,  assume  their  equation  in  the 
form 

(i)  ax^-^hy'-{-c  =  o. 


64  PROJECTIVE  GEOMETRY. 

By  the  collineation  (XIII)  this  is  transformed  into  the  equation 

(2)  (0^1+  &&1  +  cci)x+  {aa.^- bb^^ cc^,y-\-  (aa3+  bb^\cc^=o. 

This   will    be    identical   with    (i)    if   the   three   equations    hold 
aai+6&i+cCi  =  /la,  etc.,  or 

ia{a^—X)^bb^  -\-cCi  =0, 

(3)  laa^  +b(b2-X)  +  cc2  =0, 

These  are  consistent  only  if  the  determinant 
(4) 


a-  X       b, 

'2 
^3 


which  gives  a  cubic  equation  for  the  proportionality-factor  X. 
Solving  (4)  for  X  and  substituting  any  of  its  values  in  (3),  we  can 

easily  extract  the  values  of  -  and  —  from  any  two  of  equations  (3). 

c  c 

These  values  inserted  in  (i)  give  the  equation  of  an  invariant 
straight  Hne  of  the  colHneation.  As  there  are  three  values  for  X, 
there  are  three  such  lines.     Hence  the  theorem: 

A  collineation  in  a  plane  leaves  a  triangle  invariant. 

The  vertices  of  this  triangle  are  invariant  points,  while  other 
points  of  the  sides  of  a  triangle  are  generally  transformed  into 
other  points  of  the  same  sides.  (See  ex.  6  in  §  23.)  It  may 
happen  that  two  roots  of  the  determinant  (4)  are  conjugate 
im.aginary,  so  that  the  invariant  triangle  in  this  case  has  only 
one  real  side  and  one  real  point  (above  example). 

3.  In  equations  (XIII)  both  x'  and  /  become  infinite  for  all 
points  of  the  Hne 

asX+bsy+C3=o; 

hence,  in  the  collineation,  to  this  line  corresponds  the  line  at 
infinity.     Solving  equations  (XIII)  for  x  and  y,  the  result  is 


(5)  < 


x  = 


y- 


COLLIN  RATION.  65 


[__  (^2^3—  ^3^2)^'  +  (/^3^1~  ^1^3)/  +  (<^1^2~~  ^2^1)  " 

From  this  it  is  seen  that  all  points  of  the  line  at  infinity,  x=  oo, 
y=Qc  are  transformed  into  the  line 

(6)  (02^3-  a^b^)x'  +  {ajj^-  a^b^)y'  +  (^1^2-  ^2^1)  =  o. 

4.  Suppose  a  collineation  (XIII)  has  been  applied  to  a  plane. 
Turn  the  transformed  plane  through  an  angle  0  about  the  origin 

and  translate  it  afterwards  in  the  direction  tan  0  =  —  through  a 


distance  \/a'^+b'^.     The   result  of  these   successive   transforma- 
tions of  the  original  plane  {x,  y)  is  expressed  by  the  equations 


xf' 


{a^  cos  0—  flg  sin  (j)-\-aa^x-{- 
(&i  cos  ^— ^2  sin  ^+ab^y+Ci  cos  (f>—C2  sin  (fi  +  aCg 
a^x+bay+Cs  ' 

(«!  sin  ^+^2  cos  (^+603)^+ 
^^     (&i  sin  0  +  &2  cos  0 + 6^3))'  +  Ci  sin  </>  —  Cj  cos  ^ + bc^ 
^   ~  a^^rbay^rCa 

The  angle  4>,  and  a  and  5,  can  always  be  determined  in  such  a 
manner  that 

&i  cos  0— &2  sin  0-t-a&3  =  o, 
a^  sin  0+^2  cos  (f)+baa  =  o, 
a^  cos  (fi—a^  sin  (f)+aas  =  bi  sin  0+63  cos  (ji+bb-^y 

so  that  by  this  motion  (0,  a,  &)  the  equations  assume  the  form 


x"  = 


a^x+bsy+c^' 


a^x+bay+Cs 


66  PROJECTIVE   GEOMETRY. 

P 

If  the  original  plane  is  translated  in  such  a  manner  that  x  =  Xi——, 

q 

y  =  yi ,  the  connection   between  the  (x",  y")  plane  and  the 

(^i>  ^i)  plane  will  be  of  the  form 

OiX, 

oc" 


^       a^x-^^^+Y^ 

Hence  the  theorem: 

//  two  collinear  planes  {figures)  are  given,  it  is  always  possible, 
by  proper  motions,  to  bring  both  into  a  perspective  position. 

And  as  a  corollary: 

Any  two  quadrilaterals  in  a  plane  can  always  be  moved  into  a 
perspective  position;  one  may  be  considered  as  a  perspective  of  the 
other. 

As  a  special  case  we  have: 

Any  quadrilateral  may  be  considered  as  the  perspective  of  a 
square;   and  conversely. 

On  account  of  its  practical  importance  this  proposition  will 
be  treated  graphically  in  §  27. 

§  21.   Continuous    Groups    of    Projective    Transformations. 

If  to  the  point  x',  y'  obtained  by  the  projective  transforma- 
tion 

a^x-^b^y^c^ 


(I) 


a^x^b^y-^c^ 
another  transformation  of  the  same  kind 

1^^       a3^  +  /?3/+r3 


COLLIN  RATION. 


67 


is  applied,  the  result  is 

(3) 
where 


^      A^+B,y+C^' 


^i=«i«i+/5i«2+ri«3,  B^=(^A+^A+riK  c^-^a^c^-^^^c^-^r^c^, 

A^^a^a^+^^a^^-r^a^,  B,=a^b,+^^b^+r^bs,  C^=a^c^+^^c^-\-y.c^, 
A 3= 0Csa^+ {^3(^2+73(^3,  Bs=asb,  +  l^sb2+r3K  C^=a^c^  +  ^f^+Y^c^. 

Transformation  (3)  which  changes  {x,  y)  directly  into  (x'',  y) 
is  of  the  same  form  as  (i)  and  belongs,  therefore,  to  the  totality 
of  all  projective  transformations.  For  this  reason  it  is  said  that 
all  projective  transformations  of  the  general  type  form  a  group. 
It  is  eight-termed  (achtgliederig),  since  its  general  equations 
depend  upon  eight  parameters.  In  §  20  we  saw  that  every  trans- 
formation (i)  leaves  a  triangle  invariant,  and  this  fact  is  the 
characteristic  property  of  the  general  projective  group. 

It  is  not  our  purpose  to  discuss  all  possible  projective  groups, 
and  we  shall  simply  point  out  the  most  important  ones. 

The  six-termed  linear  transformation 


(4) 


x'  =  a^x-\-h{y-{-c^, 
y'  ^^a^x-^h^yArC^ 


is  clearly  a  group.  As  it  is  contained  in  the  general  group  it  is 
called  a  six-termed  subgroup  of  (i)  and  leaves  the  Hne  at  infinity 
invariant. 

The  perspective 

a.x 


(5) 


x'  = 

y 


a^x+b^-^c^' 
a^x+b^y+c^ 


68  PROJECTIVE   GEOMETRY. 

is  a  three-termed  subgroup  leaving  a  point  (origin  x=o,  y—o) 
and  the  axis  s  of  collineation  (every  point  of  it)  invariant.  As  in 
these  examples,  it  may  be  found  that  every  projective  special 
group  leaves  a  certain  figure  invariant.  The  particular  invariant 
figure  is  characteristic  for  the  group. 

The  theory  of  continuous  groups  is  a  creation  of  Sophus  Lie  ^ 
and  is  of  the  greatest  importance  in  various  branches  of  mathe- 
matics, notably  in  the  theory  of  differential  equations. 


§  22.   The  Principle  of  Duality.^ 

A.  Two  forms  of  projectivity  have  already  been  studied  (§§  6, 
9,  lo,  13) — the  projectivity  of  pencils  and  that  of  rays.  Two 
projective  pencils  generate  a  curve  of  the  second  order;  two  pro- 
jective ranges  generate  a  curve  of  the  second  class.  In  the  first 
case  the  point  is  the  generating  element  of  the  curve;  in  the 
second  it  is  the  straight  line.  In  both  cases  the  equations  in  point- 
and  line- coordinates  are  respectively  of  the  second  degree. 

A  plane  figure  may  therefore  be  considered  either  as  a  configura- 
tion of  points  or  as  a  configuration  of  straight  lines.  This  is  the 
principle  of  duality.  Two  figures  are  called  dual  if  to  a  point  in 
one  corresponds  a  straight  line  in  the  other,  and  conversely. 
Below  is  a  scheme  of-  some  dual  figures,  which  by  the  foregoing 
statements  explains  itself. 


^  Vorlesungen  iiber  contimderliche  Grtippen.  Theorie  der  Transformations- 
gruppen. 

The  theory  of  projective  groups  has  been  worked  out  synthetically  by  Pro- 
fessor Newson  and  partly  by  the  author  himself.  See  Kansas  University  Quar- 
terly, Vol.  V,  No.  I. 

^  Historic  Note. — Poncelet  in  his  Traite,  1822,  was  the  first  geometer  who 
showed  by  his  method  of  reciprocal  polars  the  great  importance  which  dualistic 
relations  have  for  geometry.  Gergonne,  in  the  Annales  de  Mathematiques,  T. 
XVI,  1826,  stated  the  principle  of  duality  in  all  its  generality  and  independently 
of  reciprocal  polars.  Plucker  first  established  the  analytic  expression  for  duality, 
and  Steiner  gave  the  equivalent  geometric  interpretation. 


COLLINEATION. 


69 


I.  Range  of  points  on  a  straight 
line. 


2.  Curve. 

Envelope. 

3.  Triangle. 

•  C 
•A           -B 

Trilateral. 

4.  Quadrangle. 

^'         -C 

Quadrilateral. 

5.  Points  of  a  plane  (00  ^). 

6.  Point  of  a  curve. 

7.  Tangents  from  a  point  to  a 

curve. 

8.  Point  of  intersection  of  two 

straight  lines. 

9.  Intersections  of  curves. 


Pencils  of  rays  through  a  point. 


Straight  lines  of  a  plane  (qo  ^). 

Tangent  of  a  curve. 

Points  of  intersection  of  a  straight 

line  with  a  curve. 
Straight    line    connecting    two 

points. 
Common  tangents  of  curves. 


B.  Analytically  the  principle  of  duality  is  expressed  by  the  dis- 
tinction between  point-  and  line- coordinates:  x,  y  and  u,  v.  If 
the  point  with  the  coordinates  x  and  y  satisfies  the  equation 

ax-\-by-\-c  =  o, 

then  the  point  is  situated  on  the  straight  line  which  cuts  from  the 

c  c 

X-  and  ;y-axes  the  distances  —  —  and  —  r-     If  ^  and  y  are  kept 

a         b 
fixed  and  a,  b,  c,  or  u  =  —,  v  =  —  are  varied,  then  for  all  values  of 

a,  b,  c,  or  u  and  v,  which  satisfy  ax+by+c,  or  ux-]-vy+ 1=0,  a 
straight  line  is  obtained  which  passes  through  the  point  (x,  y). 
Hence 

ux-\-vy+i=o 

is  the  quation  of  the  point  (x,  y)  in  line-coordinates  u,  v.  In 
§  19  the  equations  for  a  general  colUneation  were  established  for 


70  PROJECTIVE  GEOMETRY. 

point-coordinates.      The  problem  now  is  to  do  the  same  thing 
for  Hne-coordinates.     A  straight  Hne 

(i)  ao^+hy-\-c  =  o 

by  equations  (XIII),  §  19,  is  transformed  into 

x{aa^  +  ha^  +  ca^  +  ;y  (a  6^  +  &&2  +  ch^  +  ac^  +  Jcg  +  cc^ = o, 

or 

^  ^  aci+bc2+ccs      ^ ac^i- bc2+ CC3 

Putting  ax'-i-by^  +  c  into  the  form  uxf+vy'-i- 1=0  and  (2) 
into  the  form  u'x+vy+i  =  o,  the  required  equations  of  colhnea- 
tion  in  Hne-coordinates 


(XIV) 


CjU+C^V+Cs' 

biU+b.v+bo 


v 


(^  c^u+c^v+c^ 


are  obtained.  By  this  transformation  every  straight  Hne  with  the 
coordinates  u,  v  is  transformed  into  a  straight  Hne  with  the  coor- 
dinates u',  v' . 

The  discussion  of  these  formulas  is  similar  to  that  of  (XIII) 
and  may  be  left  to  the  student.  In  (XIII)  and  (XIV)  the  ana- 
lytical expressions  for  the  dualistic  interpretation  of  coUineation 
have  been  obtained.  As  for  (XIII),  the  group-property  is  funda- 
mental for  (XIV).  In  case  of  perspective  in  line-coordinates,  a^, 
^3,  &2J  ^3  vanish  from  (XIV). 

§  23.   Exercises  and  Problems. 
I.  Show  that  the  transformation 

od  =  x  cos  ^  +  ;y  sin  ^, 
y  =  X  sin  (^— }' cos  ^, 


COLLIN  RATION.  71 

consists  of  a  rotation  through  an  angle  0  and  a  reflection  on  the 
X-axis. 

2.  What  becomes  of  the  circle  x'^-\-y^  =  r^  after  a  dilatation; 
what  is  the  ratio  between  the  enclosed  areas  before  and  after 
dilatation  ? 

3.  Investigate  the  transformation 

(xf  =  ax-\-hy, 

y  =  cx-\-  dy, 
where  ad—bc=i. 

4.  The  area  included  by  a  closed  curve  C  in  the  x^y -plane  is 

obtained  by  evaluating  /  dx'dy'.      If  we  now  transform  the  x/y'- 

plane  by  the  equations  :x/  =  0(:r,  )>),  y'=(p(x,y),  the  area  of  the 
transformed  curve  c  is 


\dx  dy     "dy  Zx)  J  ^ 


Take  now  a  linear  transformation 

xf  =ax^hy-\-c, 
y  =  dx-\-ey+f, 

then  the  area  ^'  of  a  curve  in  the  transformed  plane  expressed 
in  terms  of  the  area  A  of  the  original  curve  is 

A'  =  {ae-bd)A, 


A 


or  -r-  =  ae—hd\ 

A  ' 

i.e.,  in  a  linear  transformation  corresponding  areas  have  a  con- 
stant ratio. 

5.  Prove  that  all  points  of  the  :r;>'-plane  are  transformed  into 
a  straight  line  when  ae—bd  =  o. 

^  Picakd:  Traite  d' Analyse,  Vol.  I,  pp.  98-102. 


72 


PROJECTIVE  GEOMETRY. 


6.  Find  the  invariant  points  from  the  equations  of  a  general 
collineation.     In  these  equations  set  x  =—,  y=—,  (xf  =—,  y  =—, 

•^       c       c        ^ 

and  designate  by  /^  a  factor  of  proportionaHty ;  then 


(I) 


If  i^'>  v'}  CO  is  identical  with  (f,  f],  ^),  we  get  the  condition 


(2) 


or 


(3) 


^3?+  M+(C3->^)C  =  0, 


^1~  •^  ^1 


c^-X 


This  determinant  gives  three  values  for  X,  consequently  from 
(2)  three  sets  of  values  for  (f,  vj,  ^).  Two  values  of  A,  hence 
two  of  the  invariant  points,  may  be  imaginary,  while  the  line  join- 
ing them  is  real  (§  20,  2). 

7.  Show  that  all  motions  in  a  plane  form  a  group. 

8.  Prove  the  same  for  affinity; 

9.  For  symmetry  (central  and  orthogonal); 
ID.  Similitude. 

11.  Find  the  invariant  elements  of  an  affinity. 

12.  How  does  a  linear  equation  affect  an  hyperbola? 


§  24.    Orthographic  Projection. 

I.  In  an  orthographic  projection  two  perpendicular  planes 
of  projection  are  assumed.  One  is  in  a  horizontal  position, 
and  is  designated  by  H ;  the  other  in  a  vertical  position,  •  and  is 
designated  by  V.     Both  H  and  V  intersect  each  other   in  the 


COLLIN  RATION. 


73 


ground-line  GL  and  divide  the  whole  space  into  four  quadrants, 
or  angles,  Fig.  24.  With  respect  to  the  observer,  space  may  be 
described  as  above  or  below  H,  in  front  or  back  of  V.  The  four 
angles  are  now  numbered  as  follows : 

I  Angle:  above  H,  in  front  of  V. 
II  Angle:  above  H,  back  of  V. 

III  Angle:  below  H,  back  of  V. 

IV  Angle:  below  H,  in  front  of  V. 


Fig.  24. 


In  any  of  these  angles,  the  projections  of  a  point  are  obtained 
as  the  foot-points  of  the  perpendiculars  (projectors,  projecting 
hues)  from  these  points  to  H  and  V.  If  A  is  the  point  in  space, 
we  may  designate  the  horizontal  projection  of  A,  which  is  in 
H,  by  A^,  the  vertical  projection  of  A  by  ^"  (in  V). 

Let  the  plane  through  A  A'  and  A  A''  intersect  GL  in  A*, 
then  there  is  A"A'^=AA\  A'A'^-^-AA".  In  order  to  have  the 
representation  of  these  projections  in  one  plane  (plane  of  the 
drawing;  blackboard),  one  of  the  planes,  for  example  V,  may  be 
rotated  about  GL,  so  that  the  part  of  V  above   GL  turns  from 


74 


PROJECTIVE  GEOMETRY. 


the  observer  till  it  coincides  with  H.  After  the  rotation  the 
upper  portion  of  V  covers  the  back  part  of  H,  and  the  lower 
portion  of  V  lies  in  coincidence  with  the  front  part  of  H.  Accord- 
ingly the  projections  of  points  in  the  different  angles  will  lie  as 
follows  with  respect  to  GL: 


Point  in     I  Angle 

U  a  TT  (( 

U        U      JJJ  u 


iJ-projection  below,  F-projection  above  GL 
"  above,  "  above    " 

"  above,  "  below     " 

"  below,  "  below      " 


The  same  is  true  of  the  projections  of  any  figures  situated  in 
the  different  angles. 

In  Fig.  25  these  cases  are   represented.     The  two  projections 
of  a  point  necessarily  lie  in  the  same  perpendicular  (eventually 


Fig.  25. 

extended)  to  GL.  A  fixed  point  in  space  can  be  represented  in  one 
and  only  one  way  by  two  projections.  Conversely,  two  points  in 
the  same  perpendicular  to  GL  always  represent  the  projections  0} 
a  point  (but  only  one  point). 

2.  Straight  Line. — ^A  straight  line  is  determined  by  two 
points  (the  only  line  or  curve  determined  by  two  points).  An 
orthographic  projection  (of  course,  also  a  perspective)  of  a  straight 
line  is  also  a  straight  line.  Hence,  if  A',  A'';  B' ,  B"  are  the 
projections  of  two  points,  the  lines  joining  A' ,  B'  and  A",  B" 


COLLIN  EA  TION. 


75 


Fig.  26. 


are   the   corresponding  projections   of  the  Hne  AB.     Generally 

any    straight    line — when    produced — pierces    H    and    V.     The 

points  where  this  occurs  are  called  the  traces  of  the  line  and  may 

conveniently    be    designated    by 

/j,  /j;    5^,  s^,  etc.  (^1  =  horizontal, 

/^  =  vertical  trace).     A  point  P  is 

situated  071  AB  when  P'   is  on 

A'B',       P"      on     A"B",      and 

FP"±GL,    Fig.    26.     To    find 

the   traces  t^,  /a  when  A^B'  and 

A"B^'  are  given  we  notice  that 

the  F-projection  of  a  point  in  H 

lies  in  GL,  and  that  the  -ff-pro- 

jection  of  a  point  in  V  lies  also 

in  GL;   the  other  projections  coincide  with  the  points  themselves, 

respectively. 

Hence,  to  find  the  horizontal  trace  /j  of  AB,  produce  A"B" 
till  it  intersects  GL  at  //'.  This,  being  still  situated  on  A"B" ,  is 
the  vertical  projection  of  a  point  of  AB,  and  as  it  is  also  on  GL, 
the  point  must  be  in  H  and  is  necessarily  the  required  trace  t^. 
Hence,  to  find  ty,  produce  the  perpendicular  to  GL  at  //'  till  it 
intersects  A'B'  produced  at  the  required  /j.  Similarly,  ^2  is  found 
by  drawing  a  perpendicular  at  the  point  of  intersection  t2  of 
A'B'  with  GL  and  producing  it  till  it  meets  A"B"  produced  at 
the  required  /j- 

//  is  evident  that  any  two  lines  I '  and  I "  may  he  considered  as 
the  projections  of  a  line  I.  This  line  is  uniquely  determined  hy 
I'  and  I".  To  prove  this  draw  any  two  perpendiculars  to  GL, 
cutting  I '  and  I"  2iiA',A"  and  B' ,  B" .  These  points,  however,  rep- 
resent the  projections  of  two  points  and  hence  the  problem  is 
reduced  to  the  foregoing  considerations.  This  proposition  is 
altogether  general  no  matter  how  the  lines  may  be  situated. 
//  one  line  is  perpendicular  to  GL,  the  other  reduces  to  a  point. 
Usually  the  projections  may  be  assumed  indefinitely  extended. 

A  finite  portion  may  be  cut  out  by  two  perpendiculars  to 
GL.     If  the  traces  t^  and  t^  are  given,  the  projections  are  found 


76  PROJECTIVE  GEOMETRY. 

by  drawing  the  perpendiculars  tj,^'  and  tj,.^  to  the  ground-Hne; 
then  t^t^  is  the  i?-projection,  y^'  the  F-projection  of  the  Hne. 
3.  Plane. — ^A  plane  is  determined 

{a)  By  three  points  not  in  the  same  straight  line; 

ip)         one  point  and  a  straight  line ; 

(c)         two  intersecting  hnes ; 

{d)  two  parallel  lines. 
The  most  convenient  manner  to  represent  a  plane  is  by  its 
traces,  i.e.,  its  lines  of  intersection  with  the  planes  of  projection. 
The  traces  of  a  plane  meet  in  the  grouf^d-line  and  may  be  desig- 
nated by  (7i,  02',  ■^1,  ^2;  etc.  Let  S  and  T  be  the  points  where 
a^  and  o^,  r^  and  Tj  meet.  A  plane  in  a  general  position  extends 
into  all  four  quadrants,  and  if  nothing  else  is  specified  it  will 
be  understood  that  the  plane  is  indefinitely  extended. 
Ex.  I.  Draw  the  projections  of  a  straight  Hne 

(a)  ±  H  (/)    _L  GL 

(b)  ±  V  (g)  in  a  plane  _L  GL 

(c)  II  H  (h)  cutting  GL 

(d)  II   V  (i)    in  iJ  or  F 

(e)  II  GL  ij)    in  F  and  ±  H, 

and  repeat  the  construction  in  all  four  angles;   also  construct  the 
traces  in  every  case. 

Ex.  2.  Draw  the  traces  of  a  plane 

{a)  i.  H  {e)    II  H 

(b)  JL  F  (/)    II  F 

(c)  _L  H  and  F  (g)  passing  through  GL 

(d)  II    GL  (h)  solve  the  foregoing  problems 

in  all  four  angles. 

Ex.  3.  A  profile  plane  is  a  plane  ±  GL.  Given  a  plane  ||  GL; 
find  its  distance  from  GL. 

Ex.  4.  Given  any  plane ;  locate  a  point  in  this  plane ;  i.e.,  draw 
its  projections. 

Ex.  5.  A  straight  line  lies  in  a  plane  when  its  traces  lie  in  the 
corresponding  traces  0}  the  plane;  conversely,  a  plane  passes  through 


COLLIN  RATION. 


77 


a  line  i}  its  traces  pass  through  the  corresponding  traces  of  the  line. 
Construct  the  traces  of  the  planes  determined  by  {a),  (b),  (c),  or 
(d)  under  3,  §  24. 


§  25.   Affinity  between  Horizontal   and  Vertical  Projections   of 
a  Plane  Figure. 

RABATTEMENT. 

I.  The  orthographic  projections  of  a  j&gure  in  a  plane  o^Sa^ 
are  perspectives  with  infinitely  distant  centers  in  directions  per- 
pendicular to  H  and  V.  In  this  case  any  of  the  projections  and 
the  corresponding  revolved  figure  are  in  the  relation  of  affinity 
(orthogonal  affinity  with  one  of  the  traces  as  an  axis).  But  there 
exists  also  affinity  between  horizontal  and  vertical  projections  of  a 
plane  figure,  as  we  shall  now  see. 


Fig.  27. 

2.  Let  P  be  the  bisecting  plane  of  the  first  and  third  angles, 
Q  the  bisecting  plane  of  the  second  and  fourth  angles.  The  pro- 
jections of  a  point  in  P  are  equally  distant  from  GL;  those  of  a 
point  in  Q  coincide. 

Hence,  to  find  the  point  of  intersection  of  any  line  I  with  Q, 
produce  I '  and  I "  till  they  intersect  at  X;  I  represents  the  coinciding 


78 


PROJECTIVE  GEOMETRY. 


projections  of  the  required  point  of  intersection,  Fig.  27.     If  this 
point  k  is  infinitely  distant,  i.e.,  if  /'  \\l",  then  /  is  parallel  to  Q. 

If  a  plane  P  is  given,  then  all  lines  in  this  plane  will  generally 
intersect  the  line  of  intersection  5  of  P  and  Q.  Hence  the  cor- 
responding projections  of  all  lines  in  P  will  meet  in  points  of 


s^,  s" ;  this  line  is  therefore  the  axis  of  affinity  which  exists  between 
the  projections  of  figures  in  a  plane  P.  If  ^  is  a  point  in  P 
(traces  s^,  s^,  then  we  can  rabat  P  about  s^  into  iJ.     During 


COLLIN  RATION. 


79 


If  the  hori- 


the  rabattement  A  describes  a  circle  with  center  in  ^^i  A'  describes 
a  perpendicular  to  5^.  A  Hne  /,  its  horizontal  projection  /',  and 
its  rabatted  position  I'"  meet  in  the  same  point  of  s^,  Fig.  28. 
Hence  there  exists  also  afl&nity  between  the  horizontal  projec- 
tion of  a  plane  figure  and  its  rabattement  into  H.  In  Fig.  28 
these  affinities  are  illustrated  in  case  of  a  triangle, 
zontal  projection  of  a  triangle, 
A'B'C  and  A''  are  given,  B'' 
and  C"  may  be  found  by  applying 
the  principle  of  affinity.  Thus 
A'B^  meets  A"B'^  in  a  point  r 
of  5  (/,  5"),  and  B',  B"  lie  in  a 
perpendicular  to  GL\  hence  join 
A"  with  r  and  through  B'  draw 
B'B"  LGL,  thus  determining  B'' 
Similarly  C"  may  be  obtained. 
By  the  same  principle  the  rabatte- 
ment A"'B"'C"'  may  be  con- 
structed if  one  point,  say  A'",  is 
known.  An  interesting  special  case 
is  obtained  when  s^  and  ^2  are 
equally  inclined  towards  GZ.    Then  Fig.  29. 

5  is  perpendicular  to  GL  and  corresponding  projections  of  closed 
figures  have  equal  areas,  Fig.  29.  This  case  has  been  discussed 
in  §  17,  Fig.  23,  elation. 

Ex.  I.  Discuss  the  affinity  which  results  when  s^  coincides 
with  S2. 

Ex.  2.  What  must  be  the  position  of  a  plane  P  to  obtain 
orthogonal  affinity,  §17? 

Ex.  3.  What  plane  gives  orthogonal  symmetry? 

Ex.  4.  Whal  is  the  position  of  a  plane  if  the  projections  of 
any  point  in  this  plane  are  always  equally  distant  from  the  axis  of 
athnity  si     (Oblique  symmetry.)      Make  use  of  a  profile-plane. 

Ex.  5.  A  straight  line  is  perpendicular  to  a  plane  if  its  pro- 
jections are  perpendicular  to  the  corresponding  traces  of  the  plane. 
Prove  this  proposition. 


So  PROJECTIVE  GEOMETRY. 

Ex.  6.  What  is  the  position  of  a  plane  whose  traces  coincide? 


§  26.  Homologous  Triaxigles. 

I.  In  §  15,  treating  of  central  projection,  two  planes  n  and  tz' 
intersecting  each  other  in  5  and  a  center  of  projection  V  {C)  were 
assumed.  Consider  now  any  triangle  ABC  in  n  and  find  its 
projection  A'B'C  in  tz' ;  then  a  pyramid  with  base  A'B'C  and 
vertex  V  is  obtained  which  by  the  plane  n  is  cut  in  the  triangle 

ABC.  Revolving  tz  and  with 
it  ABC  about  5  down  into  -', 
thus  assuming  the  position 
A^B^C^,  then  from  the  laws  of 
perspective  we  know  that  A'A.^, 
B'Bi,  C'C^  are  concurrent  at 
a  point  W,  and  the  points  of 
intersection  of  A'B'  and  A-^B^, 
B'C  and  B^C^,  CA'  and  C^A^ 
are  collinear,  i.e.,  lie  on  the 
same  straight  line.  Triangles 
with  this  property  are  called 
homologous,^  Fig.  30.  Any  two 
triangles  for  which  the  lines 
joining  corresponding  vertices  are  concurrent  may  always  be 
considered  as  resulting  by  the  foregoing  projection  and  rabatte- 
ment,  so  that  the  following  theorem  holds: 

//  two  triangles  A^B^C^  and  A2B2C2  are  situated  in  such  a 
manner  that  the  lines  joining  corresponding  points  like  A^A^, 
B^Bz,  C^C^  are  concurrent  at   F3,  then  the  points  of  intersectio?i 

A,B,l        B,C,\  C,A, 


---j@y^ 


Fig.  30. 


0/  corresponding  sides,  in  symbols  /t^\Tj      n  r  \^^     Ca'^'^^' 
are  collinear  on  s. 


*  Casey  in  his  sequel  to  Euclid  uses  the  word  homologous.  Mr.  Leudesdorf 
in  his  translation  of  Cremona's  Projective  Geometry,  p.  lo,  uses  homological.  See 
also  PONCELET,  loc.  cit. 


COLLIN  EA  TION. 


8i 


Conversely :  * 

//  two  triangles  are  situated  in  such  a  manner  that  the  points 
of  intersection  a,  /5,  y  of  corresponding  sides  are  collinear,  then 
the  lines  joining  corresponding  points  are  concurrent. 

It  is  noticed  that  these  theorems  are  simply  a  specialization 
of  the  general  laws  of  central  projection  or  perspective.  They 
include  evidently  the  laws  of  affinity  as  special  cases  (infinite 
point  of  concurrence,  plane  intersections  of  triangular  prisms, 
orthographic  and  generally  parallel  projections). 

2.  Theorem. — The  centers  of  homology  of  three  homologous 
triangles  with  the  same  axis  of  homology  are  collinear. 

Let  A^B^Ci,  A2B2C2,  A3-B3C3  be  the  three  triangles  whose 
corresponding  sides  meet  in  the  collinear  points  a,  ^,  y.  Con- 
sider the  two  triangles,  Fig.  31,  A^A2A^  and  3^323^,  then  it  is 


Fig.  31. 


seen  that  the  lines  A^B^,  A2B2,  A^B^  are  concurrent  at  7-,  Hence 
the  intersections  of  their  corresponding  sides  are  collinear;  but 
these  points,  V^,  V2,  F3,  are  the  centers  of  homology  of  the  given 
three  triangles,  q.e.d. 

Corollary. — The   three   triangles  A^A2A^,   B^B2B^,    C^C2C^ 


82 


PROJECTIVE  GEOMETRY. 


have  the  same  axis  of  perspective;  and  their  centers  of  homology 
are  the  points  a,  /5,  ;-.  Hence  the  centers  of  homology  of  these 
triangles  lie  on  the  axis  of  homology  of  the  triangles  A  iBjCi,  A2B2C2, 
A3B3C3,  and  conversely. 

3.  Theorem. — The  three  axes  of  homology  of  three  homologous 
triangles  with  the  same  center  of  homology  are  concurrent. 

Let  A^BjCi,  A2B2C2,  A3B3C3  be  the  three  triangles  with  the 
common   center   of   homology    V,    Fig.    32.     Consider  the   two 


triangles  formed  by  the  two  systems  of  lines  A^B^,  A  ^21  ^3-^3 
and  ^iCi,  ^2^25  ^^C^.  These  two  triangles  C3C1C2  and 
C3'Cl'C2'  are  in  perspective,  the  line  YA^A2A3  being  their  axis 
of  homology.  Hence  the  lines  joining  their  corresponding  vertices 
are  concurrent  at  S,  which  proves  the  theorem.^ 


^  See  Casey's  Sequel  to  Euclid,  ed.  1900,  pp.  77-8 
exclusively  upon  metrical  properties. 


Casey's  proofs  are  based 


COLLIN  EATION. 


83 


4.  Theorem. — If  in  two  complete  quadrilaterals  -five  pairs  of 
corresponding  sides  intersect  in  five  collinear  points,  the  point  of 
intersection  of  the  sixth  pair  will  he  collinear  with  these. 

Suppose  that  AB,  A'B';  BC,  B'C]  CD,  CD';  DA,  D'A'; 
BD,  B'D'  are  the  pairs  of  sides  intersecting  on  a  fixed  line  s.  Fig. 


33.  Now^5C  and  A'B'C  are  homologous  triangles,  consequently 
AA' ,  BB' ,  CC  pass  through  a  fixed  point  S.  Also  BCD  and 
B'C'D'  are  homologous,  and  as  BB'  and  CC  pass  through  5, 
DD'  also  passes  through  S.  Hence,  as  AA' ,  CC,  DD'  pass 
through  S,  the  triangles  ACD  and  A' CD'  are  also  homologous. 
Now  AD  and  A'D',  CD  and  CD'  intersect  in  points  of  s. 
Consequently  also  AC  and  A'C  intersect  on  s.  This,  however, 
is  the  sixth  pair,  q.e.d.  The  line  of  collinearity  may,  of  course, 
be  infinitely  distant. 

Ex.  I.  Prove  dualistically :  If,  in  two  complete  quadrangles, 
lines  joining  five  pairs  of  corresponding  points  are  concurrent, 
the  line  joining  the  sixth  pair  is  concurrent  with  these. 

Ex.  2.  Consider  any  three  spheres  in  space  which  do  not 
intersect  and  exclude  each  other.    Let  c^,  c^,  Cg  be  their  centers 


84 


PROJECTIVE  GEOMETRY. 


and  construct  their  external  common  tangent-cones  with  the 
vertices  E^,  E^,  E^.  A  common  tangent-plane  to  the  cones  {E)^, 
(£2)  is  necessarily  also  tangent  to  (E3);  hence  {E^,  (E^),  (£3) 
have  the  same  two  external  common  tangent-planes  and  E^, 
E2,  E3  are  therefore  necessarily  collinear  with  the  line  of  inter- 
section of  these  two  planes.  Similarly  it  is  seen  that  the  internal 
common  tangent-cones  of  (£1)  and  (£2)  and  of  (E^)  and  (£3) 
have  two  common  tangent-planes  which  are  also  common  to 
the  external  tangent-cone  of  (£1)  and  (£3).  Hence,  designating 
the  vertices  of  the  internal  common  tangent-cones  by  I^,  Jg,  /g, 
we  have  the  following  triads  of  coUinear  points: 

£,£2£3,    £i£2/3,    E,E,I„    E,EJ,; 

i.e  ,  the  points  0}  similitude  of  three  spheres  in  space  form  a  com- 
plete quadrilateral. 


Fig.  34. 

In  any  orthographic  projection  the  spheres  are  projected  into 
circles  and  the  £'s  and  /'s  into  their  centers  of  similitude;  and 
since  any  three  circles  in  a  plane  may  always  be  considered  as 
the  projections  of  three  spheres  in  space  excluding  one  another, 


COLLIN  RATION. 


8S 


the  foregoing  proposition  also  holds  for  three  circles  in  a  plane, 

Fig-  34. 

Ex..  3.  If  we  now  take  four  spheres  in  space  with  the  centers 
Cj,  C2,  C3,  C4,  and  designating  the  external  and  internal  centers 
of  similitude  respectively  by  E^-,  Iik  for  the  spheres  with  the 
centers  Ci  and  Ck,  then  we  find  that  all  external  centers  of  similitude 
lie  in  a  plane.  To  prove  this,  remark  that  there  are  six  centers 
of  this  kind,  E^^i  ^231  -^34>  -S41)  -^isj  ^24;  ^nd  that  these  are  arranged 
in  groups  of  three  on  six  straight  lines;  they  form,  therefore,  a 
complete  quadrilateral  and  are-  coplanar.  Fig.  35.     The  centers 


Fig.  35- 

of  similitude  of  three  spheres  are  always  coplanar ;  and  since  four 
groups  of  three  out  of  four  spheres  may  be  formed,  four  more 
quadrilaterals  of  points  of  simihtude  may  be  formed.  The 
twelve  points  of  simihtude  are  distributed  three  by  three  on  six- 
teen axes  of  simihtude.  Of  the  latter,  four  pass  through  every 
point  of  simihtude. 


86  PROJECTIVE  GEOMETRY. 

If  pqrs  designates  each  arrangement  of  the  numbers  1234, 
{pq)  the  external,  {pq)  the  internal  point  of  simiHtude  of  Cp 
and  Cq;  moreover,  if  {pqr)  is  the  axis  of  similitude  passing_through 
{pq),  ipr),  (qr),  finally  (pqr)  the  one  through  (pq),  (pr),  (qr), 
then  the  axes  may  be  represented  by  the  following  table: 


(234) 

(134) 

(124) 

(123) 

(134) 

(234) 

(123) 

(124) 

(124) 

(123) 

(234) 

(134) 

(123) 

(124) 

(134) 

(234) 

In  this  table  two  axes  which  belong  neither  to  one  and  the 
same  line,  nor  to  the  same  column,  have  always  a  point  of  simili- 
tude in  common,  while  this  is  not  true  of  two  axes  belonging 
to  the  same  column  or  line. 

This  configuration,  which  was  discovered  by  Poncelet,  is  now 
known  as  Reye's  configuration.^ 

§  27.   A  Few  Applications  to  Perspective. 

I.  Perspective  of  a  Square. — In  §  20  it  was  seen  that 
there  is  always  a  collineation  transforming  a  quadrilateral  into 
any  other  quadrilateral.  The  proof  of  this  proposition  was 
analytical.  In  view  of  its  practical  application  the  special  case 
is  of  interest: 

Every  quadrilateral  may  he  considered  as  the  central  projec- 
tion or  perspective  of  a  rectangle  or  of  a  square. 

Let  A'B'C'D'  be  any  quadrilateral  and  L'M'N'  its  diagonal 
points,  Fig.  36.  If  this  is  the  central  projection  of  a  rectangle, 
the  line  joining  M'  with  N'  must  be  the  vanishing  line  5',  and 
M'  and  N'  are  the  vanishing  points  of  the  two  pairs  of  parallel 
sides.  From  this  it  is  clear  that  the  center  of  perspective  joined 
with  M'  and  N'  gives  two  perpendicular  lines.  In  other  words, 
the  center  is  situated  on  a  circle  having  M'N'  as  a  diameter 

^  See  Archiv  )ur  Mathematik  und  Physik,  3d  series,  Vol.  I,  pp.  124-132. 


COLLIN  RATION. 


87 


(see  §  15).  Any  point  on  this  circle  as  a  center  and  any  line 
II  q'  as  an  axis  determine  a  perspective  in  which  the  original 
quadrilateral  is  the  perspective  of  a  rectangle.  This  rectangle 
can  be  constructed  without  difficulty.  The  quadrilateral  may 
also  be  considered  as  the  perspective  of  a  square.  The  center  O 
must  now  also  be  situated  on  a  circle  over  E'F'  as  a  diameter. 
O  is  therefore  the  point,  or  one  of  the  points,  of  intersection  of 


Fig.  36. 


the  circles  over  M'M'  and  E'F'  as  diameters.  In  §  8  the  har- 
monic properties  of  the  complete  quadrilateral  were  obtained 
analytically.  Constructing  the  diagonals  of  a  square,  of  which 
one  is  at  an  infinite  distance,  those  properties  appear  immedi- 
ately from  the  square,  and  as  a  projection,  does  not  change  a 
cross-ratio,  it  is  evident  that  the  same  harmonic  properties  hold 
for  the  complete  quadrilateral. 


88 


PROJECTIVE  GEOMETRY. 


2.  Perspective  of  Circles. — Every  perspective  of  a  circle 
is  called  a  conic  section  or  simply  conic.  If  two  lines  are  tangent 
to  each  other  at  a  point  A,  then  the  perspectives  of  these  lines 
are  tangent  at  the  perspective  A'  of  A.  Hence  if  a  circle  is 
inscribed  to  a  square,  the  perspective  will  give  a  conic  inscribed 
to  a  quadrilateral. 

The  problem  of  drawing  perspectives  of  circles  may  therefore 
he  reduced  to  the  problem  of  inscribing  conies  to  quadrilaterals. 

By  this  method  the  problem  can  be  solved  in  a  simpler  man- 
ner than  by  the  ordinary  construction  from  the  given  circle  and 
Qi33B32iP    the  data  of  perspective.      In  Fig.  37  a 
square  PQRS  and  the   inscribed   circle 
with   the   points   of   tangency  AA^   and 
BBi   have   been   assumed.     Divide    OB 
5  A  and  BQ   and  BP  into  the   same  num- 
bers of  equal  parts   and  number  them 
from  O  to  B  and  from  P  and  Q    towards 
B,    starting   every   time   with   o.      Con- 
nect A   with  any  of  the  division-points 
on  BP,  and  A^  with  the  corresponding 


Fig.  37. 


point  on  OB.     The  point  of  intersection  K  of  these  two  rays  is 

Q 


Fig.  38. 
a  point  of  the  circle.     In  a  similar  manner  the  points  of  the 
circle   in  the   remaining  quadrants  may  be   located.     Now  the 


COLLIN  RATION.  89 

rays  from  A  and  A^  form  two  projective  pencils,  and  their  prod- 
uct is  therefore  a  curve  of  the  second  order.  As  ZA^KA  is  a 
right  angle,  this  curve  is  a  circle  (indeed  AAiOi=  AAPi,  hence 
A11A.A1)  and  is  therefore  identical  with  the  assumed  circle. 

Now,  in  order  to  inscribe  a  conic  into  the  quadrilateral  PQRS, 
touching  at  AA^  BB^,  Fig.  38,  construct  the  point  O  as  the  inter- 
section of  the  diagonals  PR  and  QS.  Joining  O  to  M  and  N 
and  producing  gives  AA^BB^.  Applying  the  same  principle  of 
bisection  by  diagonals  in  analogy  with  Fig.  37,  the  proper  division 
on  OB  and  PQ  is  obtained.  Having  these,  the  inscribed  conic 
is  found  in  exactly  the  same  manner  as  the  inscribed  circle. 
The  proof  of  this  construction  is  evident,  since  every  quadrilateral 
may  be  considered  as  the  perspective  of  a  square,  and  the  per- 
spective does  not  destroy  the  projectivity  of  pencils. 

This  construction  is  very  effective  in  perspective  drawing, 
being  apphcable  to  all  kinds  of  quadrilaterals. 

It  is  also  remarked  that  conies  {perspectives  of  circles)  are 
curves  0}  the  second  order. 

This  idea  will  be  fully  discussed  in  the  next  chapter. 


90 


PROJECTIVE  GEOMETRY. 


§  28.   Exercises  and  Problems. 

I.  Given  two  straight  lines  and  a  point;  to  draw  a  straight 
line  through  this  point  passing  through  the  inaccessible  point  of 
intersection  of  the  given  lines. 

Solution. — Let  g  and  h  be  the  given  hnes  and  A  the  given 
point.  Through  A  draw  any  two  Hnes  cutting  g  and  h  in  PQ 
and  RS,  Fig.  39a  and  395.  join  PS  and  QR  and  produce  till 
they  intersect  at  M.     Designating  the  inaccessible  point  by  N, 


Fig.  39a. 


PQRS  may  be  considered  as  the  perspective  of  a  square  having 
M,  N,  A  as  diagonal  points.  Hence  any  third  line  through  M 
cutting  g  and  h  at  T  and  V  is  the  perspective  of  a  line  parallel  to 
QR  and  PS.  From  this  it  follows  that  the  line  joining  A  to  the 
point  of  intersection  B  of  TR  and  QV  is  the  required  line. 

2.  Inscribe  a  conic  within  a  rectangle;  within  a  trapezium;  a 
rhombus. 

3.  Draw  two  homologous  quadrilaterals. 


COLLIN  RATION. 


91 


FiG.  39&. 


4.  Draw  the  perspective  of  a  cube;  of  a  cylinder;  of  a  pyramid; 
of  a  hexagonal  prism. 

5.  Draw  the  perspective  of  two  concentric  circles. 


CHAPTER  III. 

THEORY  OF  CONICS. 

§  29.    Introduction. 

The  Greeks  originally  studied  conies  as  plane  sections  of 
cones. ^  S.teiner  and  Chaslcs  considered  them  as  products  of 
projective  pencils  and  ranges,  defined  by  anharmonic  ratios, 
von  Staudt  and  Reye,  however,  define  this  relation  purely  by 
harmonic  division.  I  shall  follow  Steiner's  method,  by  which 
the  projective  properties  of  the  circle  (see  §  12)  are  easily  ob- 
tained and  transformed  to  conies  by  central  projection.  Follow- 
ing this  method,  it  becomes  necessar}''  to  show  that  all  curves  of 
the  second  degree  as  obtained  by  projective  pencils  and  ranges 
are  also  produced  by  plane  sections  of  cones,  or  as  perspective 
coUineations  of  the  circle. 

Conversely,  it  must  be  shown  that  every  curve  of  the  second 
degree  may  be  projected  into  a  circle.  This  is  the  method  followed 
by  Poncelet,  Steiner,  and  a  majority  of  modern  writers  on  pro- 
jective geometry.  From  a  purely  geometrical  standpoint  von 
Staudt's  and  Reye's  methods  are  to  be  preferred. 


1  Menachmtjs  obtained  conies  as  intersections  of  planes  perpendicular  to  the 
elements  of  a  right  cone.  In  case  of  an  "acute-angled  cone"  (opening  angle  at  the 
vertex  <  90°)  the  conic  was  called  ellipse;  of  a  "right-angled  cone"  (angle  at 
vertex  =90"^)  a  parabola;  of  an  ""obtuse -angled  cone""  (angle  at  vertex  >  90°)  an 
hyperbola.  Apollonius,  who  introduced  these  names,  extended  the  proofs  to 
oblique  cones. 

92 


THEORY   OF   CONICS.  93 


§  30.   Identity  of  Curves  of  the  Second  Order  and  Class  and 

Conies. 

The  general  equation  of  a  circle  is 

(i)  {x-ay+{y-hy  =  r\ 

To   obtain   the   equation    of   this    circle   in   line- coordinates, 
assume  the  equation  of  its  tangent  in  the  form 

(2)  ux-\-vy^i=o. 

This  represents  a  tangent  if  the  distance  of  the  center  (a,  h) 
from  the  line  (2)  is  r;   i.e.,  if* 

au  bv  I 


\/u2+V2        'VU2+V2        \^U2-\-V^ 


or 


(3)  r^(u^-\-v-)—  {au+bv+iy  =  o. 

Every  pair  of  values  u,  v  satisfying  (3)  gives  the  line-coor- 
dinates of  a  straight  line  tangent  to  the  circle  (i).  Equation  (3) 
represents,  therefore,  (i)  in  line-coordinates  (see  §  6).  Both  (i) 
and  (3)  depend  upon  three  essential  parameters.  The  formulas 
for  perspective  in  their  dual  interpretation  each  depend  upon 
three  essential  parameters.  Hence,  applying  a  perspective  to 
either  (i)  or  (3),  i.e.,  to  the  given  circle,  we  can  in  both  cases 
'  dispose  of  the  six  parameters  in  such  a  manner  that  the  trans- 
formed equations  assume  any  given  form  of  the  second  degree 
in  X  and  y,  or  in  u  and  v.  This  means  that  every  curve  of  the 
second  order  or  class  may  he  considered  as  a  conic  section  {per- 
spective of  a  circle). 

Conversely,  if  the  general  equation  of  a  conic  of  the  second 
degree  is  given,  it  is  always  possible  to  find  a  perspective  which 
will  transform  this  equation  into  that  of  a  circle.  Hence  every 
curve  of  the  second  degree  may  he  projected  perspectively  into  a 
circle.     Curves  of  the  second  degree  arid  conies  are  therefore  identical. 


94  PROJECTIVE  GEOMETRY. 

Ex.  I.  The  perspective  transformation 


Ix+my+n^ 


y= 

"^      Ix+my+n 

transforms  the  general  equation 

ax'''^-\-hy'^+  cody' -\-  2dx' -{-  2ey' -\-  j  =  o 
into 

x^{a+  2dl-\-  jP)  +  y"{h+  2em^  jm'^)  +  xy{c-\-  2(11^1+  2el+  2Jlm)-\- 

x(2dn+  2Jln)  +  y(2en+  2Jmn)  +  /w^ 

Find  the  values  of  /  and  m  which  will  transform  the  given 

equation  into  that  of  a  circle. 

Ex.  2.  Solve  the  dual  problem  of  Ex.  i. 

Ex.  3.  Find  a  circle  and  a  perspective,  so  that  the  perspec- 

x"^     y"^ 
tive  of  the  circle  is  the  ellipse  -^  +  -7^=  i. 

Ex.  4.    Given   the    circle   x'^^y'^=\.     Transform   this    circle 
by  the  perspective 


y-i'    '    /-I 

Discuss  the  result  geometrically  and  show  that  the  center  of 
the  circle  is  the  focus  of  the  transformed  circle. 


§31.   Linear  Transformation  of  a  Curve  of  the  Second  Order. 
I.  By  the  translation 

(I)  1^  =  /  +  ^ 


-      THEORY   OF  CONICS.  95 

the  equation 
(2)  ax'^-{-2hxy-\-cy'^-\-2dx+2ey+j  =  Q 

is  transformed  into 

ax''^-\-  2hx' y -{- c' y^  +  2(aa^+hhi-\-d)x'  +  2(6^1+  c&i+  e)/  + 

aa^  +  2haJ).^^+  ch^ -\-  2da.^-]-  2eh^-\-  j  =  0. 

In  order  that  the  coefficients  of  x'  and  y'  disappear,  a^  and  h^ 
must  be  chosen,  so  that 

aai-\-hhi-\-d  =  o, 
If  ac— &2  5^o,  we  find  for  a^  and  61  the  values 


(3) 


he—cd 


,    K= 


bd—ae 
ac  —  h'^' 


By  this  assumption  the  transformed  equation  reduces  to  (the  o<f 
and  y  being  replaced  by  x  and  y) 


(4) 
where 


ax'^-{-2hxy-\-cy'^-^ 


ac—b^ 


=  0, 


A  =  d{be-  cd)  +  e(&(^-  ag)  +  j{ac-  b^) 


a  b  d 
bee 
d  e  j 


2.  Turning  the  coordinate  axes  now  through  an  angle  6;  i.e., 
making  the  transformation 


(5) 


\x=x  cos  d—y  sm  d, 
\  ^  =  x^  sin  d+y  cos  d, 


and  again,  suppressing  the  primes  of  x  and  y,  the  result  is 


Ax''-\-2Bxy^Cy''+ 


A 


ac-b""    ^' 


96  PROJECTIVE  GEOMETRY. 

where 

A^a  cos^  d-\-  2b  sin  d  cos  (9+  c  sin^  ^, 
2B=  (c—a)  sin  2(9+ 2&  cos  26, 
C  =  a  sin^  d—2b  sin  ^  cos  ^+  c  cos^  ^. 

Choosing  d  so  that  5  =  o,  or 

2b 

(6)  tan  2^  = , 

the  transformed  equation  reduces  to 

(7)  Ax^+Cy'=T, . 

^'^  -^       o-—ac 

To  determine  A  and  C,  we  have  from  the  foregoing  expressions 
for^,  25,  C: 

(       ^+C  =  a+c, 
•^^^  (52-X-C  =  &2-ac. 

But  when  6  is  chosen  so  that  B  =  o,  A  •C  =  ac—b^.     Hence  A  and 
C  are  roots  of  the  equation 

z^—  (a+c)z+ac— &^=o. 

If  now  &^—acF^o,  two  cases,  ¥—ac>o  and&^— ac<o,  must  be 
distinguished. 

It  is  further  assumed  that  J  5^0.     In  the  first  case,  b^—ac>o, 

it  follows,  since  now  AC  =  ac—b^<  o,  that  A  and  C  are  of  different 

i 

sign.       Hence,  no   matter  what   the  sign  of  J,    -ttt; ^  and 

^  '  o  '    A(b^—ac) 

■TTTT^ ^  have  different  signs  and  are  always  real.      The  equa- 

C(b^—ac) 

tion  therefore  represents  an  hyperbola.    If,  in  addition  to  &^—  ac  >o, 

J  =  o,  then  the  equation  may  be  resolved  into  two  linear  factors; 

the  hyperbola  degenerates  into  two  intersecting  straight  lines. 

In  the  second  case,  b^  —  ac<o,A-C  =  ac-b^>o-     Both  A  and 


THEORY   OF    CONICS.  97 

^  A 

C  have  the  same  sim;  hence  also  777^ r  and  :^T7ro r  have 

*   '  A(b^—ac)  C{P—ac) 

the  same  sign.  According  as  this  is  positive  or  negative,  the 
equation  represents  a  real  or  an  imaginary  ellipse.  If  J  =  o,  the 
ellipse  degenerates  into  two  intersecting  imaginary  lines. 

3.  Finally  the  case  b^—ac  =  o  must  be  considered.  Here 
b=±\/ac.  Considering  the  case  &  =  +\/ac,  the  general  equa- 
tion (2)  reduces  to  (coordinates  oc',  y') 

(9)  {^ac<f  +  \^cy'Y-\-2do(f-\-2ey'-\-j==o. 

Putting  \^a  =  r  sini9,  \/c  =  r  cos  (9;  i.e.,  r'^  =  a-\-c,  tan  ^  =  v-> 
and  dividing  the  whole  equation  by  r"^,  we  have 

(10)  {o(f  sin  (9  +  /  cos  Oy-{-^  (2dx'+2ey'  +  f)=o. 

Turning  the  coordinate  axes  through  an  angle  d;   i.e.,  putting 

x  =  x^  cosd—y'  sind, 
y  =  x' sind  +  y' cosO, 

or  ^=x  cos /9+>' sin  ^, 

y  =  —  x  smd  +  y  cos  d, 

equation  (10)  becomes,  after  putting  sin  6=     .,  cos  6=    . -, 

Va-\-c  Va+c 

and  reducing, 

2  (         -  - 

(11)  y^+ -,=  <{dVc—eVa)x+ 

(a+c)Va+c( 

{dV a+  eV c)y-{- ^jV a-\- c  [  =0. 

Making  finally  the  translation,  by  replacing  y  and  x  by  y+^ 
and  x+a  respectively,  we  have 


98 


PROJECTIVE   GEOMETRY. 


(a+c)va+c  (^  -^ 

— (a+c)\/fl+c+i/Va+c  r. 


Letting 


(12)  becomes 


^      e\/c-[-d\/a  , 

^y  = and 

■    (a+c)\/a+c 

(g\/^+^\/a)2+/(a+c)2 


«  =  — 


2  ((fVc— eV  a)  (a  +  c)Va+ c ' 
dy/c—eVa 


-y'=_2- 


((Z+c)Va+c 
which  represents  a  parabola.     When  b^—ac  =  o,  then 


J  = 


bee 
d  e  f 


—  (cd^—  2bde-{-  ae^). 


But  &=  +Vac,  hence 

J  ==  —  (ae^—  2  V  ac  •  de+cd^)  =  —  (<?Va—  (/Vc)' 


hence 

(13) 


r2C. 


(a+c)\/<z+c- 

If  in  (11)  d\^c—e\/a  =  o,  i.e.,  J=o,  then  the  equation  may  be 
resolved  into  two  linear  factors  in  y  only,  and  represents  con- 
sequently two  parallel  lines.  These  are  real  and  distinct,  coinci- 
dent (real),  imaginary  and  distinct,  according  as 

The  case  b  =  —\^ac  may  be  treated  in  a  similar  manner  and 
leads  to  no  new  result. 

4.  From  the  foregoing  short  discussion  it  is  seen  that  the 
character  of  the  general  equation  of  the  second  order  may  be 
established  by  means  of  translations  and  rotations;  i.e.,  by  special 


THEORY  OF  CONICS. 


99 


cases  of  the  linear  transformation.     In  this,  two  algebraic  expres- 
sions between  the  coefficients  are  of  fundamental  importance, 

he. 

,  which  for  abbreviation  we  may  designate 


namely,  b^—ac  = 
by  T,  and 


a  b 


A  =  d{be-  cd)  +  e{bd-  ae)  +  j{ac-  P)  ■■ 


a  b  d 
bee 
d  e  f 


> 


According  as  t  =  o  the  general  equation  represents  an  hyper- 
bola, a  parabola,  or  an  ellipse,  if  J 9^0.  When  J=o  these 
curves  degenerate  into  intersecting,  parallel  or  coincident,  or 
imaginary  intersecting  lines.     The  determinant  J  is  called  the 

h       n 

DISCRIMINANT  of  the  equation.      We  may  call  x  =  h2—ac  = 

the  characteristic  determinant,  or  simply  characteristic. 

We  shall  now  show  that  the  discriminant  and  characteristie 
0}  an  equation  of  the  second  order  are  not  changed  by  a  translation. 
By  the  translation  in  which  x  and  y  are  replaced  by  x-{-a^  and 
;y-f  &i,  equation  (2)  is  transformed  into 
(14)  a.v'+  2hxy-{-cy'^-{-  2{aa^-\- 66^+  d)x+  2{ba^-\- cb^-\- e)y 

+  a^{aai-[-hhi+  d)  +  hi{ba^+  cb^+  e)  +  da^+  ebi-\-f  =  o. 
From  this  it  is  apparent  that  z  has  remained  invariant.     The 
discriminant 

a  b  aa^+bbi+d 

b  c  ba^+cb^+e 

-bbj^+d     &fli+c&i-t-e     ai(aai+bb^-\-d)-{-bi{bai-{-cbi+e) 

+  dai+ebi+f 

a  b  d 

bee 

aa^+bbi+d        ba^  +  cb^+e         dai+eb^+f 

a      b       d 

bee 

d      e      / 

which  is  the  original  discriminant. 


PROJECTIVE   GEOMETRY. 


In  a  translation  t  and  A  are  invariants.  The  same  is  true 
of  T  and  J  in  a  rotation,  and  consequently  in  any  motion  of  the 
plane  in  itself.  No  special  proof  for  the  case  of  rotation  will 
be  given,  since  it  is  contained  in  the  linear  transformation  which 
will  now  be  treated. 

5.  A  linear  transformation  (§  19,  XII) 


(15) 


y=joc'-\-dy'  +  r), 


leaves  the  line  at  infinity  invariant.  It  may  therefore  be  ex- 
pected that  such  a  transformation  does  not  materially  change 
the  expressions  t  and  A. 

As  the  constants  ^  and  t)  mean  simply  a  translation  in  addi- 
tion to  the  special  linear  transformation 


(16) 


y=Yx'  +  dy , 


and  as  a  translation  does  not  change  t  and  A,  it  is  sufficient  to 
study  the  effect  of  (16)  upon  the  general  equation  (2).  Making 
in  (2)  for  X  and  y  the  substitution  (16)  and  afterwards  replacing 
x/  and  y  by  x  and  y,  we  have 

(17)       {aa'+2har^cf)x-'+2\aa^-\-h{ad+^r)  +  (^r^\^y 

-i-(al^^+2bl^d+cd^)y^+2{da+er)x+2(d^+ed)y+}  =  o. 

The  discriminant  of  this  equation  is 


J' 


aa^+2bar+cf         aa^+h{ad-^^r)  +  <^r^    da+ey 
aal^+b{ad+^r)  +  (^r^         a^'+2b^d+cd^  d^+ed 

da+ey  d^+ed  / 


which  reduces  down  to 


(18) 


A'  =  {ad-^ry 


a  b  d 

bee 
d  e  } 


=  {ad-^ry^' 


THEORY   OF   COMICS.  lOl 

In  a  similar  manner  there  is  found 

(19)  T'  =  (ao^-/?r)^r. 

From  this  it  follows  that  the  character  of  a  curve  of  the  second 
order  is  not  changed  hy  a  linear  transformation. 

It   is   always   assumed   that   ad—^y^o.     Indeed,    according 

asT  =  o,  also  1^  =  0;  and  also  as  J  =  o,  J  =0. 

For  a  rotation,  /  =  (cos^  ^+sin"  ^)z-  =  t, 

J'  =  J; 

i.e.,  in  this  case  t  and  J  are  invariants.  In  the  next  section  it 
will  be  seen  that  conies  are  characterized  by  their  pole-  and 
polar  involutions  on  the  Kne  at  infinity  (involution  of  diam- 
eters). 

Thus,  it  is  also  geometrically  evident  that  in  a  linear  trans- 
formation, which  does  not  change  the  involution  of  the  infinitely 
distant  line,  the  type  of  a  conic  is  not  changed. 

Ex.  I.  Find  (18)  from  the  preceding  unsolved  determinant. 
Calculate  also  /. 

Ex.  2.  If  F^—ac  =  o,  assume  b  =  —Vac  and  transform,  with 
this  condition,  the  general  equation  (2)  to  the  normal  form 
y^  =  2px. 

Ex.  3.  Discuss  the  curve  determined  by  y^~  2xy-rX^—i=^o. 

Ex.  4.  WTiat  curve  does  the  equation 

ax^+(a+b)xy-\-by^-\-(a+c)x+(b  +  c)y+c  =  o 

represent  ? 

Ex.  5.  If  in  the  hnear  transformation  ad—^Y  =  o,  i.e.,  -7i"^~i 

p     0 

we  have 


x^^i^x'  +  y^, 


I02  PROJECTIVE  GEOMETRY. 

X       8 
Hence,  no  matter  what  the  values  of  x'  and  y'  may  be,  —  =  -^  = 

constant.     The  whole  x'y'-plsint  is  transformed  into  the  straight 
line  xd—y^  =  o. 


§  32.   Polar  Involution  of  Conies.     Center.     Diameters.     Axes. 

Asymptotes. 

I.  In  §§  12  and  13  the  involutoric  properties  of  the  circle 
have  been  explained.  As  a  collineation  does  not  change  pro- 
jective properties,  it  is  clear  that  the  following  theorems  hold 
for  conies.  (As  the  figures  of  involution  referred  to  in  this  sec- 
tion are  in  close  analogy  with  those  of  the  circle,  their  reproduc- 
tion is  left  to  the  student.) 

I.  The  polar s  of  the  points  0}  a  straight  line  I  pass  through 
a  fixed  point  L,  the  pole  of  I. 

II.  The  poles  of  the  rays  of  a  pencil  P  lie  on  a  straight  line  p, 
the  polar  of  P. 

From  this  follows  immediately 

III.  //  the  pole  L  of  a  straight  line  I  lies  on  a  second  line  g, 
then  the  pole  G  of  g  lies  on  I. 

IV.  //  the  polar  I  of  a  point  L  passes  through  the  pole  G  of  a 
second  line  g,  then  the  line  g  passes  through  the  pole  of  I. 

Consider  now  any  straight  line  /  and  its  pole  L.  Let  a  be 
any  ray  through  L  cutting  /  at  ^.  The  pole  A^  oi  a  lies  on  I. 
Hence  the  pole  of  the  ray  a^,  passing  through  L  and  A^,  coin- 
cides with  A.  Taking  any  number  of  rays  a,  b,  c,  d,  .  .  .  and 
constructing  as  before  their  corresponding  rays  a^,  h^,  q,  d^,  .  .  .  , 
a  system  of  coincident  polars  and  poles  through  L  and  on  /  are 
obtained  which  are  in  involution;  i.e.,  every  pair  of  correspond- 
ing rays  and  poles,  a-x^  and  XX.^,  are  harmonic  with  the  double- 
rays  and  double-points  through  L  and  on  /,  respectively.  The 
double-elements  are  real  when  /  intersects  the  conic  really;  i.e., 
when  L  admits  of  two  real  tangents  to  the  conic.  If  I  does  not 
intersect  in  real  points,  then  the  involution  has  imaginary  double- 
elements.     In  case  that  /  is  tangent  to  the  conic,  the  double- 


THEORY  OF   CON  I  OS.  103 

elements  are  coincident.  The  points  of  intersection  of  /  and 
the  tangents  from  L  to  the  conic,  whether  real,  coincident,  or 
imaginary,  give  in  all  cases  the  double-elements  of  the  involu- 
tion. Accordingly,  hyperbolic,  parabolic,  and  elliptic  involu- 
tions are  distinguished. 

Two  corresponding  rays  x,  x^,  and  I,  with  their  poles  X,  X^,  and 
L  always  form  a  self- polar  triangle;  i.e.,  a  triangle  zvhose  vertices 
are  the  poles  of  its  opposite  sides. 

All  these  properties  might  be  derived  directly  from  the  gener- 
ation of  conies  by  projective  pencils  and  ranges;  i.e.,  without 
reference  to  the  perspective  of  the  circle.  We  have  used  this 
method  to  lay  particular  stress  upon  the  invariance  of  these 
properties  by  projective  transformations.  Examples  will  be  given 
later  on  to  show  how  some  of  the  propositions  (all,  for  that  matter) 
in  this  connection  may  be  derived  independently  of  perspective. 

2.  It  is  now  of  the  greatest  interest  to  investigate  the  involu- 
tions of  poles  and  polars  when  the  latter  are  assumed  in  special 
positions.  Let  /  be  at  an  infinite  distance.  Then  for  every  ray 
a  through  L  cutting  the  conic  at  P  and  P^  (LAPPj)  =  —  i  = 
(L  CO  PPt),  or  LP=  —LP^.  Every  ray  through  L,  the  pole  of  the 
line  at  infinity,  therefore  cuts  the  conic  in  two  points  which  are 
equally  distant  from  L.  This  point  is  therefore  called  the  center 
of  the  conic. 

To  every  ray  through  the  center  corresponds  an  infinitely  dis- 
tant pole.  Call  the  center  O.  Of  great  importance  is  the  polar 
involution  through  O.  The  poles  A  and  A^  of  any  two  corre- 
sponding rays  through  O  are  infinitely  distant.  Any  ray  through 
A  cutting  «!  in  B  and  the  conic  in  C  and  D  is  parallel  to  a,  and 
(BACD)  =  {B^  CD)  =  -  I ;  hence  BC=-BD;  i.e.,  B  bisects  CD. 
Two  corresponding  rays  of  the  involution  through  O  are  called 

CONJUGATE  DIAMETERS  of  the  COuic. 

Including  imaginary  elements,  the  foregoing  properties  of  the 
polar  involution  at  the  center  give  the  following  theorems: 

V.  All  chords  of  a  conic  parallel  to  a  diameter  are  bisected  by  its 
conjugate  diameter.  The  relation  between  conjugate  diameters  is 
reversible. 


I04  PROJECTIVE   GEOMETRY. 

VI.  //  a  diameter  intersects  a  conic,  then  the  tangents  at  the 
points  of  interseciion  are  parallel  to  the  conjugate  diameter. 

3.  Definitions. — The  rectangular  pair  of  the  polar  involution 
at  the  center  are  called  the  axes  of  the  conic. 

The  double-rays  of  the  involution  of  diameters  of  a  conic  are 
called  the  asymptotes. 

The  points  at  which  the  polar  involution  is  rectangular  are 
called  the  foci  of  the  conic. 

In  these  definitions  it  is  assumed  that  the  involutions  exist. 
From  the  definition  of  a  self-polar  triangle  it  is  easily  concluded 
that  the  involutions  on  two  of  its  sides  are  hyperbolic,  while  on  the 
third  it  is  elliptic.  If  we  now  consider  a  self-polar  triangle  hav- 
ing the  center  O  of  the  conic  as  one  of  its  vertices,  then  two  dis- 
tinct cases  may  occur. 

First.  The  involutions  of  poles  on  two  conjugate  diameters  may 
both  be  hyperbolic,  while  the  polar  involution  at  the  center,  con- 
sequently the  involution  of  poles  on  the  line  at  infinity,  is  elliptic. 
The  involution  of  conjugate  diameters  has  no  real  double-rays;  the 
conic  is  an  ellipse. 

Second.  The  involution  at  the  center  is  hyperbolic;  it  is  hyper- 
bolic on  one  diameter  and  elliptic  on  the  conjugate  diameter.  The 
involution  of  conjugate  diameters  has  real  double-rays;  i.e.,  the 
conic  has  real  asymptotes  and  is  an  hyperbola. 

From  theorem  V  it  follows  immediately  that  the  ellipse  and 
hyperbola  are  symmetrical  with  respect  to  both  axes. 

The  existence  of  the  ellipse,  hyperbola,  and  a  special  involu- 
tion of  diameters  in  connection  with  a  conic,  called  parabola,  will 
be  proved  in  the  next  section.  At  the  same  time  the  existence 
of  foci  will  be  proved. 

§  33.   Existence    of    Ellipse,    Hyperbola,    Parabola,    and    their 

Foci. 

I.  In  Fig.  40,  just  as  in  Fig.  19,  §  15,  let  .y  be  the  axis,  q'  and 
r  the  counter-axes,  and  C  the  center  of  collineation.  Assume  any 
circle  K  with  C  as  a  center  and  determine  the  pole  O  of  r  with 


THEORY  OF  CONICS. 


105 


respect  to  K.  Draw  also  the  polar  involution  at  O,  of  which  OM 
and  OMi  are  a  corresponding  pair  intersecting  K  aX  X,  Y  and 
Xi,  Fi,  respectively.  Now  {OMXY)  =  {OM^Xj;)  =  -i.  In 
the  central  projection  r  and  consequently  M  and  M^  are  projected 


Fig.  40. 


to  infinity.  Hence,  designating  the  projected  figure  by  primes, 
(0'ooZ'FO  =  (0'ooX/F/)  =  -i;  i.e.,  0'X'  =  -0'Y',  OOY/-  - 
0'Y^\  O'  is  the  center  of  the  transformed  circle,  and  the  polar 
involution  at  O'  becomes  the  involution  of  diameters.  Now  the 
polar  involution  at  O  is  elhptic,  parabolic,  or  hyperbolic  accord- 
ing as  r  does  not  cut  K,  touches  K,  or  intersects  it  in  two  points. 
The  same  is  evidently  true  of  the  involution  of  diameters  at  O' . 
In  case  that  r  is  tangent  to  X,  O  coincides  with  the  point  of  tan- 
gency  of  f ;  O'  is  projected  to  infinity,  so  that  the  diameters  be- 
come all  parallel.  This  is  the  case  of  the  parabola.  A  parabola 
may  therefore  be  considered  as  a  conic  tangent  to  the  line  at 
infinity.  With  the  existence  of  these  different  involutions  of 
diameters  the  existence  of  the  ellipse,  the  parabola,  and  the  hyper- 
bola is  proved.     To  sum  up,  the  central  projection  oj  a  circle  is  an 


io6  PROJECTIVE   GEOMETRY. 

ellipse,  a  parabola,  or  an  hyperbola  according  as  it  does  not  intersect 
r,  is  tangent  to  r,  intersects  r.  Further,  a  conic  is  an  ellipse, 
a  circle,  a  parabola,  or  an  hyperbola  according  as  the  involution  of 
diameters  is  elliptic,  elliptic  and  rectangular,  parabolic  (parallel 
diameters  with  infinite  center),  or  hyperbolic. 

2.  Consider  now  the  rectangular  polar  involution  at  the  center 
O  of  the  circle  K.  The  central  projection  of  O  coincides  with 
itself,  and  for  the  corresponding  pair  of  two  perpendicular  diam- 
eters AB  and  DE,  A'B'1.D'E'.  The  points  of  intersection  P  and 
Q  of  DE  and  AB  with  q'  are  the  poles  of  A'B'  and  D'E.  The 
vanishing  line  q'  is  therefore  the  polar  of  C  with  regard  to  K' . 
The  same  holds  for  any  pair  of  perpendicular  diameters  of  K  and 
their  transformations. 

The  involution  of  polars  at  C  of  K'  is  therefore  rectangular; 
C  is  a  focus  of  K'. 

Any  conic  which  is  the  perspective  of  a  circle  with  the  center 
of  perspective  as  a  center  has  this  center  as  a  focus. 

The  construction  also  shows  that  a  focus  lies  on  the  major  axis 
of  the  conic. 

Ellipse  and  hyperbola  are  symmetrical  with  respect  to  their 
axes;  both  curves  have  therefore  two  foci  (real).  That  a  conic 
cannot  have  more  than  two  real  foci  is  seen  from  the  construc- 
tion of  Fig.  40,  and  also  from  the  fact  that  every  point  not  on 
the  axes  admits  of  oblique  pairs  of  polars. 

The  double-rays  of  the  rectangular  polar  involutions  at 
the  foci  pass  through  the  circular  points  at  infinity;  they  may 
be  considered  as  imaginary  tangents  to  the  conic  from  its 
foci.  The  foci  of  a  conic  may  therefore  also  be  defined  as 
follows : 

The  foci  of  a  conic  are  the  points  of  intersection  of  the  tangents 
from  the  circular  points  at  infinity  to  the  conic. 

Two  of  these  intersection-points  are  real,  the  other  two  are 
imaginary  and,  on  account  of  the  symmetry,  are  necessarily  situ- 
ated on  the  other  axis.  Ellipse  and  hyperbola  admit,  therefore, 
also  of  two  imaginary  foci. 

Analytically  this  also   appears  by  writing  the  equations  of 


THEORY   OF   CONICS.  107 

the  imaginary  double-rays  at  the  foci,  when  their  distance  from 
the  center  of  the  conic  is  c: 


I. 
2. 

3- 

4- 

(x—c)  +  iy  =  o. 
(x—c)  —  iy  =  o. 
{x+c)  +  iy  =  o. 
{x-{-c)  —  iy  =  o. 

From  I  and  2, 
From  3  and  4, 
From  I  and  4, 
From  2  and  3, 

x  =  c;  y  =  o. 
x=  —  c;  y  =  o. 
x  =  o;  y=  —ic. 
x  =  o',  y=  -\-ic. 

The  solutions  of  i  and  3,  2  and  4  give  the  circular  points. 

§  34.    Construction  of  Foci  Independent  of  Central  Projection. 

To  the  pencil  T  of  parallel  rays  a,  b,  c,  .  .  .  considered  as 
polars  of  the  conic  K,  Fig.  41,  correspond  the  poles  A' ,  B' ,  C,  .  .  . 
on  the  conjugate  diameter  n  of  the  direction  of  T.  To  the  diam- 
eter m  II  T  corresponds  as  pole  the  infinitely  distant  point  of  n. 
To  the  line  at  infinity  belonging  to  the  pencil  T  corresponds 
as  pole  the  center  /'  of  K. 

From-^',  B' ,  C,  .  .  .  ,  /',  draw  rays  a',  h' ,  c',  .  .  .  ,  /'  per- 
pendicular to  m.  These  rays  form  another  pencil  5  of  parallel 
rays  which  is  projective  to  the  pencil  T.  As  these  two  pencils 
are  perpendicular,  their  intersection  A^,  B^,  Q,  .  .  .  ,  M^  is  an 
equilateral  hyperbola,  having  m  and  its  perpendicular  through 
r  as  asymptotes.  Both  pencils  T  and  S  intersect  the  axes  each 
in  two  coincident  projective  ranges,  for  instance 

{ABC  ...)7\{A^B^C.^...). 

On  this  axis,  to  the  point  /'  of  the  first  range  corresponds 
the  infinitely  distant  one  on  the  same  axis.  If  F  is  taken  as  a 
point  of  the  second  range,  then  its  corresponding  one  is  infinitely 
distant.      Hence  /'  and  the  point  at  infinity  on  the  horizontal 


lo8  PROJECTIVE  GEOMETRY. 

axis  may  be  interchanged  without  disturbing  the  projectivity ; 
the  foregoing  point-ranges  form  therefore  an  involution  (§  3). 
The  double-points  of  this  involution  are  the  points  where  the 
equilateral  hyperbola  H  cuts  the  axis.  To  the  ray  /  (pencil  T) 
through  one  of  these  points,  say  F,  corresponds  the  perpendicular 
ray  /'  in  the  pencil  5.     But  every  ray  of  5  passes  through  the 


Fig.  41. 


pole  of  the  corresponding  ray  in  T.  Hence  /'  contains  the  pole 
of  /,  and  /  and  j'  form  consequently  a  rectangular  pair  of  the 
polar  involution  around  F.  The  polar  involution  of  any  point 
on  an  axis  contains,  however,  another  rectangular  pair,  namely, 
the  axis  itself  and  the  perpendicular  to  it  through  the  given  point. 
The  polar  involution  about  F  has  therefore  two  rectangular 
pairs  and  is  consequently  itself  rectangular  (§  5).  The  point 
F,  according  to  definition,  is  therefore  a  focus.  As  there  are 
two   double-points   of   the   involution    (^5C  ...)  77(^2-^2^2  ••  •)? 


THEORY   OF   CONICS.  109 

there  are  also  two  foci.  Assuming  that  the  points  of  a  pair  AA^, 
are  on  the  same  side  of  the  center  of  involution,  then 

rA-I'A^=  +  k\ 

and  FF  =  k ;  the  foci  are  real.  But  on  the  other  axis  /M*  •  FA  ^  = 
—  k~,  i.e.,  the  double-points  of  the  involution,  or  the  foci,  are  imag- 
inary. This  is  in  accordance  ^^ith  the  statement  in  the  foregoing 
section.^ 

Ex.  I.  Carry  out  construction  of  this  section  on  a  large  sheet, 
Ex.  2.  Instead  of  taking  an  ellipse  for  the  conic  R,  take  an 
hyperbola. 

§  35.    Focal  Properties  of  Conies. 

I.  From  Fig.  40,  §  33,  we  have,  if  R  designates  the  point  of 
intersection  of  PD  with  r  (not  shown  in  the  figure), 

{CocDR)  =  {CPD'^^), 

or  CD:CR  =  CD'  -.PD'. 

Designating  the  distances  of  C  and  D'  from  r  and  q'  by  y  and  ^, 
respectively,  there  is 

CR:PD'  =  r-^  =  CD:CD'; 
consequently 

CD'     CD 

-^  = =  constant. 

0  r 

This  result  may  be  stated  by  the  theorem: 

The  ratio  between  the  distance  of  any  point  of  a  conic  from  one 
of  its  foci  and  the  distance  of  the  point  from  the  polar  of  this  focus  is 
constant. 

^  Since  an  involution  of  right  angles  does  not  admit  of  real  double-rays,  it 
follows  that  the  foci  are  within  the  conic;  i.e.,  within  that  portion  of  the  plane 
from  which  no  real  tangents  may  be  drawn.  They  are  situated  on  an  axis,  since 
in  any  other  case  the  polar  involution  would  have  obhque  pairs. 


no 


PROJECTIVE   GEOMETRY. 


Definition. — The  polar  of  a  focus  of  a  conic  is  called  directrix. 

This  theorem,  as  well  as  its  converse,  may  be  used  to  define 
conies,  as  was  done  by  Pappus  (Mathematical  Collections) : 

The  locus  of  a  point  whose  distances  from  a  'fixed  point  and  a 
fi-xed  straight  line  (not  passing  through  the  fixed  point)  have  a  con- 
stant ratio  is  a  conic.  The  fixed  point  is  the  focus,  the  fixed  line 
the  corresponding  directrix. 

CD 

If,  in  Fig.  40,  K  does  not  intersect  r,  i.e.,  if <i,  K'  is  an 

CD 

ellipse ;   if  K  touches  r,  —  =  i ,  and  K'  is  a  parabola ;   if  K  inter- 
CD 
sects  r,  >  I ,  and  K'  is  an  hyperbola.     The  figure  may  easily 

be  drawn  for  the  case  of  an  hyperbola  or  a  parabola. 


Fig.  42 


2.  From  Pappus'  metrical  definition  a  number  of  properties 
of  conies  may  be  derived.  Given  a  conic  K  and  its  foci  F  and  F^, 
Figs.  42  and  43,  according  as  we  take  for  K  an  ellipse  or  an  hy- 


THEORY    OF  CONICS. 


perbola.     Both  curves  are  symmetrical  with  respect  to  both  axes. 

The  ratios  ( )  are  therefore  the  same  for  both  foci  and  their 

corresponding  polars  (directrices).  Taking  any  point  ^  on  i^  and 
designating  the  focal  distances  AF  and  AF^^  by  r  and  r^,  and  the 
distances  from  the  corresponding  directrices  by  d  and  d^,  we  have 


-T  =;r  =  constant  (Pappus). 


From  this 


r±t\ 
d±d^' 


=  same  constant  as  above.     But  in  an  ellipse 


\\ 

\    \ 

\\ 

c 

^""i~" 

1     ^k 
v/7j 

\ 

\ 

B.--''' 

7/ ' 

r 

A 

W/ 

JF 

N 

^'^ 

j\~ 

/ 

\\ 

7/ 

// 
// 

■^ 

f      V 

\ 

Fig.  43. 

(/+ (ij  =  constant,  and  in  an  hyperbola  (f—c^i  =  constant.  Hence 
the  theorem: 

The  sum  of  the  radii  vector es  (AF,  AF^)  0}  any  point  0}  an 
ellipse  is  constant. 

The  difference  of  the  radii  vectores  of  an  hyperbola  is  constant. 
In  both  cases  the  constant  is  equal  to  tlfe  distances  of  the  vertices  of 
the  curves. 


PROJECTIVE   GEOMETRY. 


The  second  part  of  this  theorem  results  by  taking  A  in  one 
of  the  vertices.     In  case  of  an  eUipse  we  have  then 


in  case  of  an  hyperbola 


r 

d' 


r+r^- 


■r+^d,; 


r-r^  =  r--d,. 

3.  Given  a  conic  K,  Fig.  44,  and  its  foci  F  and  F^.  Take 
any  point  P  in  the  plane  of  K  and  construct  the  polar  involution 
around  P  and  its  rectangular  pair  PR^,  PR.  Connecting  P 
with  all  pairs  of  the  involution  on  the  axis  which  are  formed  by 
couples  of  rectangular  conjugate  polars  of  K  parallel  to   PR 


Fig.  44. 

and  PRi  (see  Fig.  41,  §  34),  an  involution  of  rays  at  P  is  obtained 
in  which  PF  and  PF'  are  the  double-rays,- and  PR,  PR^  the 
rectangular  pair.  PR  and  PR^  are  consequently  the  bisectors 
of  the  angles  formed  by  PF  and  PF^.  In  the  polar  involution 
around  P,  the  tangents  PU  and  PV  from  P  to  K  are  the  double- 
rays  and,  according  to  the  construction,  PR,  PRy  the  rectangular 
pair.  The  angles  formed'  hy  PU  and  PV  are  therefore  also 
bisected  by  PR  and  PR^.     Hence  the  theorem: 


THEORY  OF  CONICS. 


113 


The  angles  formed  by  the  tangents  {real)  from  a  point  to  a  conic 
are  bisected  by  the  bisectors  of  the  rays  joining  this  point  to  the 
foci;  or  these  tangents  form  equal  angles  with  the  focal  rays  (PF, 
PF,). 

If  P  lies  on  K,  say  at  Q,  then  PU  and  PV  coincide  with  the 
tangent  t  at  Q.     We  have  therefore  the  corollary: 

The  tangent  at  any  point  of  a  conic  includes  equal  angles  with 
the  focal  radii  at  this  point. 


Fig.  45. 


4.  If  in  Figs.  42  and  43  F^A  is  produced  and  AB  made  equal 
to  AF  {F^B  =  2a,  the  major  axis),  then  BF±AC  (tangent  at  ^), 
hence  BC=FC.     As  OC  \\F.^B  =  ^F^B,  we  have  the  theorems- 

The  locus  of  the  reflected  images  of  a  focus  on  all  tangents 
of  an  ellipse  or  an  hyperbola  is  a  circle  having  the  other  focus  as 
a  center  and  the  major  axis  as  a  radius. 

The  locus  of  the  foot-points  of  all  perpendiculars  from  the  foci 


114 


PROJECTIVE  GEOMETRY. 


of  an  ellipse,  or  an  hyperbola,  to  their  tangents  is  a  circle  over  the 
major  axis  as  a  diameter. 

In  case  of  a  parabola,  Fig.  45,  the  first  circle  becomes  the 
directrix  /^  and  the  second  the  tangent  v  at  the  vertex.  Suppose 
P  to  be  a  point  where  two  perpendicular  tangents  of  the  parabola 
meet,  and  let  A  and  A'  be  the  points  of  tangency.     We  have 

a  +  a:'=— ;    hence  AF^  and  A'F-^^  include  an  angle  of  n.     D  is 

therefore  the  pole  of  a  focal  chord  A  A'  and,  as  such,  lies  in  the 
directrix.     Hence : 

The  tangents  from  any  point  of  the  directrix  of  a  parabola  to 
the  parabola  are  perpendicular  to  each  other. 

It  is  known  from  §  33  that  the  polar  involutions  around  a 
focus  are  rectangular.  Thus,  if  AA'  is  a  focal  chord  and  A^  its 
pole  on  the  directrix,  FA^1,AA'  at  F.  The  foregoing  statement 
is  therefore  only  a  part  of  the  general  proposition,  since  /.AFA^  = 
right  angle,  Fig.  46: 


Fig.  46. 


The  portion  of  a  tangent  of  a  conic  between  its  point  of  tangency 
and  its  intersection  with  the  corresponding  directrix  appears  as 
subtended  by  a  right  angle  when  seen  from  the  focus. 


THEORY  OF  CONICS.  115 

Further,  from  the  fact  that  the  polar  of  a  point  which  is  situated 
on  another  polar  passes  through  the  pole  of  this  polar,  and  that 
the  polar  involution  at  a  focus  is  rectangular  it  follows.  Fig.  46: 

The  rays  joining  a  focus  with  the  point  of  intersection  of  two 
tangents  and  the  point  of  intersection  of  the  chord  of  contact  of 
these  tangents  with  the  corresponding  directrix  are  perpendicular. 

Ex.  I.  If  from  any  foint  O  at  a  distance  c  from  the  center 
of  a  circle  with  radius  r  two  perpendicular  secants  are  drawn, 
intersecting  the  circle  in  the  points  A,  B,  C,  D,  then 

Assuming  this  proposition,  prove: 

The  locus  of  the  point  of  intersection  of  two  tangents  to  an 
ellipse  or  an  hyperbola  which  cut  at  right  angles  is  a  concentric 
circle.  If  a  and  b  are  the  parameters  (major  and  minor  half- 
axes),  the  radius  of  the  circle  is  y/a^±b^. 

Ex.  2.  Let  Q  and  R  be  the  points  of  intersection  of  a  third 
tangent  with  the  two  tangents  to  a  parabola  from  a  point  P. 
Prove  that  ZRPF  is  equal  to  the  angle  which  QP  makes  with 
the  diameter  of  the  parabola  through  P. 

Ex.  3.  Applying  the  proposition  estabHshed  in  the  fore- 
going exercise,  prove  that  the  circle  circumscribing  a  triangle 
formed  by  three  tangents  to  a  parabola  passes  through  the  focus. 

Ex.  4.  The  locus  of  the  foci  of  all  parabolas  which  touch  the 
three  sides  of  a  given  triangle  is  the  circumscribing  circle  of  the 
triangle.     (Cremona.) 

Ex.  5.  Suppose  a  quadrilateral  A  BCD  is  circumscribed 
about  a  conic  with  the  points  of  tangency  KLMN,  Fig.  47.  The 
pairs  of  sides  ^5  and  CD,  BC  and  AD,  CA  and  BD  intersect 
each  other  in  the  three  points  OPQ.  The  pole  of  ^C  is  the 
point  of  intersection  of  KN  and  LM  and  is  the  point  of  con- 
currence of  PO,  ML,  DB,  NK.  Similarly,  KL,  AC,  NM  meet 
•at  a  point  of  PO,  the  pole  of  BD.  Hence,  in  a  quadrilateral 
circumscribed  to  a  conic,  the  diagonals  form  a  self-polar  tri- 
angle.    If  .4,  B,  N,  and  L  are  given,  then  we  may  choose  Q 


ii6 


PROJECTIVE   GEOMETRY. 


Fig.  47. 


THEORY   OF  CONICS. 


117 


at  random  on  NL,  by  which  the  fourth  side  CD  and  its  point 
of  contact  is  perfectly  determined.  From  this  the  proposition 
follows : 

Let  PAB  he  a  triangle  circumscribed  to  a  conic,  and  LN  the 
chord  of  contact  of  the  tangents  PB  and  PA ;  then  the  lines  join- 
ing any  point  Q  on  LN  io  A   and  B  are  conjugate  polars. 

Ex.  6.  Prove:  The  portion  of  a  movable  tangent  of  a  central 
conic  between  the  two  tangents  at  the  vertices  subtends  right  angles 
at  the  foci. 

Ex.  7.  The  lines  joining  the  points  of  intersection  of  all  circles 
through  the  foci  with  the  tangents  at  the  vertices  of  a  central  coriic 
are  the  tangents  of  this  conic.  (The  two  points  joined  must  not 
be  equally  distant  from  the  axis.) 

Ex.  8.  Consider  again,  Fig.  48,  two  tangents  intersecting 
at  P,  and  let  their  chord  of  contact  AB  intersect  the  directrix 


Fig.  48. 


at  C.  Then  PF  is  the  polar  of  C.  Consequently  FAPBC  is 
a  harmonic  pencil  in  which  one  pair,  FP,  FC  is  perpendicular. 
FP  therefore  bisects  the  angle  AFB,  §  5.  Hence  the  proposi- 
tion : 

The  line  joining  a  focus  to  the  point  of  intersection  of  two  tan- 


Il8  PROJECTIVE   GEOMETRY. 

gents  of  a  conic  bisects  the  angles  formed  by  the  rays  joining  the 
focus  to  the  points  of  contact. 

Ex.  9.  Consider  two  fixed  tangents  and  a  movable  tangent 
of  a  conic,  and  join  a  focus  to  their  points  of  intersection  and 
points  of  contact.  Applying  the  proposition  of  Ex.  8,  prove  that 
the  piece  of  a  movable  tangent  included  by  two  fixed  tangents  appears 
under  a  constant  angle  from  a  focus. 

In  case  of  an  hyperbola  whose  asymptotes  include  an  angle  (f>, 
the  above  constant  angle  with  reference  to  the  asymptotes  as 

fixed  tangents  is  ;r . 

°  2 

Ex.  ID.  The  extremities  of  a  tangent,  determined  by  the  asymp- 
totes, and  the  foci  of  the  hyperbola  are  concyclic. 

By  means  of  this  proposition  it  is  easy  to  construct  an  hyper- 
bola by  its  tangents  when  the  asymptotes  and  the  foci  are  known. 


§  36.   Analytical  Expression  for  Tangent  and  Polar. 

I.  Although  problems  connected  with  tangents  and  polars 
of  general  conies  have  so  far  been  simply  treated  without  their 
analytic  forms,  it  is  of  great  value  for  the  developments  that  will 
follow  to  establish  their  equations  by  means  of  transversals  and 
anharmonic  ratios.  Let  (x^,  y^),  {Xr^,  y^  be  two  points  A  and  C; 
then  the  coordinates  of  any  point  B  of  the  straight  line  A  C  are 
given  by  the  equations 

x^-Xx^  y-ly.^  .    AB 

Substituting  these  values  in  the  general  equation 

w  =  ax^+  2bxy-\-  cy^-{-  2dx-\-  2ey-\-f  =  o, 

and  multiplying  by  (i  —  Xy,  we  obtain,  after  arranging  according 
to  ascending  powers  of  A, 


THEORY  OF  CONICS.  II9 

(2)  U^—2Xv+X^U2  =  o, 

where 

u^  =  ax^^  -\-  2hx.j^  +  cy^^  -\-  2dxi-\r  2ey^-\-  f, 
u^  =  ax^^  2bx^y^-^cy^^  2dx^-^  2ey^-\-], 
V  =  Xi(ax2-\-  by2+  d)  +  y^ihx^^  cy^-^  e)  +  (/X2+  e>'2+  / 
=  ^2  (ax,  +  hy^  +  (^)  +  :V2  (^^1 + <^yi  +  ^)  +  dx^  +  ey^  +  /. 

From  (2)  two  values  of  X  are  obtained  which  when  substituted 
in  (i)  give  the  coordinates  of  the  points  of  intersection  of  J.C 
with  the  conic  U .  In  case  that  these  two  points  coincide,  the 
roots  of  (2)  will  be  equal  and^C  is  a  tangent  to  V.  Now,  the 
condition  for  equal  roots  is 

(3)  V^-U,U2  =  c,. 

Suppose  that  {x^,  ^'2)  itself  is  on  U,  then  ^2=0,  and  the  con- 
dition reduces  to  v  =  o.  Every  point  (x,,  y^)  which  satisfies  v  =  o 
lies  on  the  tangent  at  {x^,  ^'2).  The  equation  of  the  tangent  at 
(^2,  y^  is  therefore 

(4)  x{ax2+hy2+d)  +  y{hx2+cy2^e)  +  dx2+ey2+j  =  o. 

2.  Let  A  and  B  be  the  points  (x,,  y^),  {x^,  y^)  and  C,  D  the 
points  of  intersection  of  the  straight  line  AB  with  C/,  correspond- 
ing to  the  roots  of  (2).     li  A,  B  and  C,  D  are  two  harmonic 

•       .^.       ^A^,r^■n^  •        ACAD  AC     AD 

pairs,  then  (^5CiP)  =  -  I ;  I.e.,  ^:-g^=- I,  or -^^-^-^^=0. 

^      AC  AD 

But  -^  and  ^   are  the  roots  of  (2);   hence  A  BCD  form 

V 

a  harmonic  group  if  the  sum  of  the  roots  of  (2);   i.e.,  2  —  =  0. 
As  W2  is  not  supposed  to  be  on  U,  this  is  only  possible  if  2;=o;  i.e.,  if 

(5)  x^iaxj^  +  by^  +d)  +  y2{hx^ + cy^  -}-  e)  +  dx^  +  g^^ + /  =  o. 


I20  PROJECTIVE  GEOMETRY. 

If  four  points  ABCD,  of  which  C  and X>  are  on  TJ,  are  colhnear 
and  form  a  harmonic  group,  then  the  coordinates  of  A  and  B  are 
related  by  (5).  Keeping  the  point  A  fixed  and  letting  B  under 
condition  (5)  vary,  it  is  plain  that  all  points  B  restrained  by  these 
conditions  lie  on  the  straight  Hne 

(6)  x{ax^  +  hy^  +  ^)  +  y{hx^  +  cy^  +  e)  +  dx^  +ey^-\-j  =  o. 

This  line  is  called  the  polar  of  the  point  A  {x^,  y^  with  respect 
to  U. 

Similarly,  the  polar  of  B(x2,  3^2)  is  given  by 

(7)  x{ax2+  by2+  d)  ^y{hx^^^  cy^-^  e)  +  dx^-]-  ey^+f  =  o. 

If  (X2,  y2)  hes  on  the  polar  (6)  of  (x^,  y^),  then  (5)  holds.  But 
this  can  also  be  written  as 

(8)  x,(ax2+  hy^^+d)  +  yi{hx^+  cy^^  e)  +  dx^+  ey^\  /  =  0, 

which  is  the  condition  that  (x^,  y^  lies  on  the  polar  of  (Xg,  y^. 
Hence  the  theorem  w^hich  has  already  been  established  before : 

//  A  is  on  the  polar  0}  B,  then  B  is  on  the  polar  of  A. 

From  this  it  follows  that  the  polars  of  the  points  of  a  straight 
hne  all  pass  through  its  pole,  and,  conversely,  the  poles  of  all  rays 
through  a  fixed  point  he  on  its  polar. 

3.  To  establish  the  relation  between  the  points  of  a  straight 
line  and  the  corresponding  pencil  of  polars,  assume  first  any  four 
lines  through  a  fixed  point : 

p^^a^x+b^y+c^^o, 
p2  =  a2^+  Ky  +  ^2  =  0, 

Pi- tJ-p2  =  o. 

Cut  these  lines  by  any  transversal  and  find  the  anharmonic 
ratio  of  the  four  points  of  intersection  A^A^A^A^.  For  the  sake  of 
simplicity,  choose  the  x-axis  as  this  transversal,  so  that  the  distances 
of  these  points  from  the  origin  become 

C^  C^  Cj      /C2  Cj      11C2 


b,'        b^        b.-Xb^'        b-iih 


THEORY   OF   CONICS.  121 

The  anharmonic  ratio  is  easily  found: 

(9)  (^.^^^^.).M..M_=.i. 

If  now  {x^,  y-^,  (^2,  y^  are  the  coordinates  of  two  points  A^^ 
A  2,  then  the  coordinates  of  any  point  A^  on  AiA2  are 

Xi-Ax^      y^-ly^        -  .    A^A^ 

The  polar  of  ^3  is 


/ ,  X.— }.x^      y,— ^^2 


x,-Xx2       yi-h'2.f 


or,  multiplying  by  i  —  >^  and  rearranging, 

X  (ax^  +  by^  +d)  +  y  (bx^  +  c^i  +  e)  +  dx^  +  <?3'i  +  / 

-  X\x(ax\+  &>'2+  (/)  +  y{bx2+cy2+  e)  +  c?X2+  ^3^2+  /I  =0. 

Designating  the  equations  of  the  polars  of  A^,  A^  simply  by 
pi  =  o,  p2  =  o,  the  polar  of  A^  will  be  represented  by  p^—  kp2  =  o: 
Analogously,  the  polar  of  a  fourth  point  A^  on  A^A^  is  repre- 

A  A 

sented  by  pi—  ,up2  =  o,  where  ,« =  .    .  .     The  anharmonic  ratio  of 

A  A  A  A        •  •  ^  1  3       1  4       1         -^  V^  3    -^  2-^  3          '^  a  i 

^^.  ^2,  ^3,  ^4  IS  evidently  -r--  :T"7~  =  irT'- j~l~  =  ~     Accord- 

^3^2    -^4^2        ^1^4     -^2^4         /" 

ing  to  (9)  the  same  is  true  of  the  four  points  of  intersection  of  any 
transversal  with  the  polars  pi  =  o,  p2  =  o,  pi—  ^p2  =  o,  pi— l^p2  =  o 
of  the  points  Aj^,  A 2,  A 3,  A^.     Hence  the  theorem: 


122 


PROJECTIVE  GEOMETRY. 


The  range  of  points  0}  a  straight  line  and  the  corresponding 
pencil  of  polars  are  projective. 

This  follows  also  by  considering  the  polar  involution  around 
a  point  and  the  corresponding  involution  of  poles  on  its  polar, 
as  shown  elsewhere. 

4.  Equation  of  a  Conic  in  Line-coordinates. 

To  find  the  equation  in  line-coordinates  u,  v  oi  2,  conic  with 
the  Cartesian  equation 

(10)  ax^-\-2'bxy-\-cy'^-\-2dx-\-2ey-\-f  =  o, 
consider  the  equation  of  a  tangent 

(11)  {ax^-^ly^-^  d)x+  {hx^+  cy^+  e)y+  {dx^^  ey^+  f)  =0 

at  a  point  {x^,  y^  of  this  conic.     The  line  coordinates  of  this 
tangent  are 

ax^  +  hy^+d 


(12) 


w  = 


v  = 


dx^i-ey^+f' 
bXi-\-  cyi+  e 
dx^+ey^+f 


Conversely,  the  Cartesian  coordinates  x^,  y^  expressed  in  terms  of 
the  line-coordinates  u  and  v  of  the  tangent  at  this  point  are,  from 

(12), 

(cf-  e^)u+  (de-  bf)v+  (be-  cd) 


(13) 


Ji 


(be—  cd)u-\-  (bd—  ae)v-\-  (ac—  P) 
(de-  bf)u+  (af-  d^-)v+  (bd-  ae) 
(be—  cd)u-\-  (bd—  ae)v+  (ac—  b^) 


But  these  values  of  x^  and  y^  satisfy  (10).     Thus,  substituting 

(13)  in  (10),  we  get  the  relation  which  exists  between  the  line- 
coordinates  u  and  V  of  the  tangents  of  the  conic  (10),  or  the  equa- 
tion of  the  conic  in  line-coordinates.  After  reduction  this  equa- 
tion becomes 

(14)  (cf-  e^)u^+  2(de-  bf)uv-\-  (af-  d^)v^+ 

2  (be—  cd)u+  2  (bd—  ae)v-{-  ac—  b^=o. 


THEORY  OF  CONICS.  1 23 

Ex.  I.  Find  the  line-equations  of 


x^     y^ 

y^  =  px\ 

x-+y^-  =  r^\ 

(x- 

-fl)2+(3;-&)2-;-2  =  o. 

Ex.  2.  Establish  the  equation  of  a  point  of  (14).. 
Ex.  3.  From  the  line-equation  of  a  conic, 

aii"-\-  2buv+cv^+  2du+  2ev-\-f  =  o, 

establish  the  Cartesian  equation. 

§  37.   Theory  of  Reciprocal  Polars. 

I.  We  have  already  discussed  the  principle  of  duality,  §  22, 
in  an  elementary  manner.  In  this  section  it  will  be  seen  that 
the  principle  follows  directly  from  the  theory  of  polars. 

To  every  point  as  a  pole  corresponds  a  straight  line  as  a  polar, 
and  conversely.  To  two  projective  pencils  producing  a  conic 
correspond,  according  to  the  theorem  at  the  end  of  §  36,  3,  two 
projective  point-ranges  which  produce  a  conic  as  an  envelope; 
i.e.,  to  the  points  of  a  conic  correspond  the  tangents  of  another 
conic,  called  the  reciprocal  of  the  first.  In  general,  to  any  figure 
consisting  of  points  and  straight  lines  corresponds  a  figure  con- 
sisting of  straight  lines  and  points,  the  polars  and  poles  of  the 
points  and  lines  of  the  first  figure.  The  anharmonic  ratios  of 
corresponding  elements  are  the  same  in  the  original  and  reciprocal 
figure. 

The  transformation  thus  established  is  called  polar  reci- 
procity and  is  contained  in  the  slightly  more  general  principle 
of  duality.  The  polar  of  a  point  {x^,  y^  with  respect  to  the 
conic  U  is  given  by 

(i)  x(ax^+  by^+d)  +  y(bx^-{-cy^-{-e)  -\-  dx^+  ey^+}  =  o, 


124  PROJECTIVE  GEOMETRY. 

or,  introducing  the  line-coordinates, 

ax^-\-hy^-\rd 
,  .  .  \^~  dx,+  ey,+  f' 

^     ~  dx,+  ey,+  f' 
(3)  xu+yv-\- 1=0. 


Formulas  (2)  are  the  analytical  expression  for  this  trans- 
formation. To  the  point  (x^,  y^)  corresponds  the  straight  line 
with  the  coordinates  (u,  v).  As  the  transformation  is  involu- 
toric;  i.e.,  that  the  coincidence  of  a  point  and  straight  line  neces- 
sitates the  coincidence  of  their  polar  and  pole,  we  can  inter- 
change {x,  y)  with  (Xj,  y^),  as  has  been  already  established. 

Designate  now  the  original  conic  by  U,  the  conic  to  be  recip- 
rocated by  Ky  and  the  reciprocal  conic  by  iTj.  Assume  U  and 
Ky  as  central  conies.  If  the  center  of  U  is  outside  of  K^,  two 
tangents  from  it  may  be  drawn  to  K^,  which  when  reciprocated 
are  two  points  of  K^.  As  these  tangents  pass  through  the  cen- 
ter of  Z7,  their  poles  will  be  infinitely  distant.  From  this  it  fol- 
lows that  K2  is  an  hyperbola.  If  K^  passes  through  the  center 
of  U,  then  only  one  real  tangent  can  be  drawn  to  K^  at  this  point ; 
i.e.,  K2  will  have  only  one  infinite  point  (tangent)  and  is  there- 
fore a  parabola.  When  the  center  of  U  is  inside  of  K^,  no  real 
tangents  from  it  can  be  drawn  to  K^;  i.e.,  K2  has  no  infinite 
points  and  is  consequently  an  ellipse.     Hence  the  theorem: 

According  as  the  center  of  the  original  conic  U  is  outside,  on, 
or  inside  of  the  conic  K^  to  he  reciprocated,  the  reciprocal  conic  K2 
will  he  an  hyperhola,  a  parabola,  or  an  ellipse. 

2,  According  to  (2)  {x^,  y^  is  the  pole  of  the  line  with  the 
coordinates  {u,  v).  Suppose  now  that  this  line  envelopes  a 
circle  of  radius  r  and  having  its  center  in  the  origin  of  coordinates. 
The  line-equation  of  this  circle  is 

(4)  u'-\-v'^  =  ^. 


THEORY  OF  CONICS.  1 25 

Taking  U={x—ay+{y—^y—p'  =  o;  i.e.,  a=i,  6  =  0,  c=i, 
d=—a,  e=— /?,  /  =  a:^+/?"— ^-,  substituting  these  values  in  (2), 
and  finally  substituting  the  values  of  u  and  v,  thus  obtained  in 
(4),  we  get 


or,  expanded  and  rearranged, 

(5)  (^^~ a^)Xj^—  2al^Xiyi+  (^^— /3^))'i^+  2a(aH/5^—  p"^—  O^i 

+  2l3(a'+^'-p'-r')y,+  r'(a'+^')-(a'+l^'~-pY  =  o. 

This  is  the  equation  to  which  the  poles  of  all  tangents  of 
(4)  are  subjected.  Hence  (5)  is  the  equation  of  the  conic  recip- 
rocal to  the  circle  x^  +  y^  =  r^  with  respect  to  the  circle  (x—ay 
+  {y—^y  =  P^-     Here  the  characteristic  determinant  of  (5)  is 

Evidently  t>,  =,  <o,  according  as  a:^-j-/?^>,  =,  >y^ 
Hence,  according  as  the  center  of  U  is  outside  of,  on,  or  inside 
of  (4),  the  reciprocal  conic  (5)  is  an  hyperbola,  a  parabola,  or 
an  ellipse,  which  is  in  agreement  with  the  previous  result. 

Ex.  I.  What  is  the  reciprocal  of  a  polygon  circumscribed 
to  a  conic  with  respect  to  this  conic? 

Ex.  2.  Find  the  reciprocal  of  the  point  Au  +  Bv-\-C  =  o. 

Ex.  3.  Find    the    reciprocal    of    the    envelopes    u^—v^  =  o] 

Ex.  4.  Discuss  reciprocation  in  the  case  where  in  formulas 
(2)  the  determinant 


=  0. 


a 

b 

d 

b 

c 

e 

d 

e 

1 

126  PROJECTIVE  GEOMETRY. 

Ex.  5.  Given  the  polar-reciprocal  transformation 

Ax^By^D 
11 

Bx^Cy+E 


v= 


Dx+Ey+F' 


by  which  to  every  point  {x,  y)  corresponds  a  straight  line  {u,  v), 
and  conversely. 

Establish  the  equation  of  the  conic,  for  which  every  point 
(x,  y)  coincides  with  the  corresponding  Hne  (u,  v). 


§  38.   General  Reciprocal  Transformation.     Polar  Systems. 

I.  Formulas  (2)  of  the  foregoing  section  may  be  generalized 

by  setting 

a^x+b^y+Ci 


(I) 


u  = 


v  = 


aaX+bsy+Cs' 

a^x+b^y^c^ 
asX+b^y  +  c^' 


Solving  (i)  for  x  and  y,  we  get 

{a^bs-  a^b^u+  {a^b^-  a^b^)v+  (aj)2-  a^b^ 


y= 


To  a  straight  line 


ax^by^c  =  o 


corresponds  by  this  transformation  a  point  with  the  line-equation 


(3) 


{a(&2C3-  &3C2)  -f  b{a^C2-  a^c^  +  c(a2^3-  ajj^  ]  u+ 

\a{bsCi-  b^Cs)  +  b(a^Cs-  a^c^)  ■\-  c{a^b^-  a^b^)\v+ 

I  {a(6iC3-&2Ci)  +  &(a2Ci-aiC2)  +  cK&2-«2^i)l  =0. 


THEORY   OF  CONICS.  1 27 

Conversely,  to  a  point 

corresponds  a  line  with  the  equation 

(4)        (aa^'i-ba2+ca3)x+(abj^+bb2+cbs)y+(aCi-\'bc2+cc^)=o. 

If  four  points,  of  which  no  three  are  collinear,  with  the  equa- 
tions a^u+b'h)+c^=o  (i  =  i,  2,  3,  4)  are  given;  and  also  four 
arbitrary  lines,  of  which  no  three  are  concurrent,  with  the  equa- 
tions a^x-\-^y+f  =  o  (i  =  i,  2,  3,  4),  we  can  let  these  points  and 
hnes  correspond  to  each  other  in  a  reciprocal  transformation 
by  setting 

r%a^aj^+  ¥a2+  c^a^  -  a*(a*q-|-  &V2+  ck^  =0, 

^■=1,  2,  3,  4, 
f{a%  +  b% + c%)  -  ^\a'c^  +  bk^ + c%)  =  o. 

These   are   eight  equations  with   the  eight   unknown  ratios 

«1     «2     «3     ^1      ^2     ^3     Cj      c,  I,-   -u  .U      1    ..  u      r 

7-5  ~j  ~>  ~j  ~>  ">  ~j  ^j  irom  which  the  latter  may  be  found 

^3        ^3        ^3        ^3        ''S        ''3        ^3        ''3 

definitely.  Hence  the  theorem:  A  quadrilateral  and  a  quad- 
rangle always  determine  a  reciprocal  transformation  in  which 
they  correspond  to  each  other.  The  reciprocal  transformation  is 
the  most  general  duahstic  transformation  and  includes  polar 
reciprocity  as  a  special  case.  This  is  easily  recognized  by 
comparing  formulas  (2)  of  §  37  and  (i)  of  this  section. 

2.  We  shall  now  determine  those  Hnes  of  the  coincident  planes 
{u,  v)  and  {x,  y)  which  coincide  with  their  corresponding  points. 
A  line  with  the  coordinates  u  and  v  passes  through  the  point  with 
the  coordinates  x  and  y  if 

ux -{- vy -\- 1  =  o. 


128  PROJECTIVE  GEOMETRY. 

Hence  the  points  {x,  y)  whose  corresponding  Hnes  {u,  v)  accord- 
ing to  (i)  pass  through  them  satisfy  the  condition 


a^x  +  h^y  +  C3       a^  +  hy + c{ 


or 


(5)  <iiX^+  {h^-\-a^xy+\y^^  (Ci+  03)^+  {c^+h)y+c^^<^, 

which  represents  a  real  or  imaginary  conic  C.  Conversely,  for  the 
lines  (u,  v)  whose  corresponding  points  {x,  y)  lie  on  them,  we  have 
the  condition 

(6)  (V3-  ^3^2)^'+  (&3C1-  ^1^3+  «3C2- ^2^3)^^+  KC3-  ^3^1)^'+ 
(^^^3—  &2C1+  02^3—  ajo^u-\-  {a2C^—  a^C2-{-  ajb^—  afi^v+ 

which  represents  a  conic  of  the  second  class  F.  The  conies  C 
and  r  are  generally  different,  as  may  be  seen  by  applying  the 
results  of  §  36,  4,  to  equations  (5)  and  (6). 

To  every  point  of  C  correspond  the  two  tangents  from  it  to 
-T;  conversely,  to  every  tangent  of  F  correspond  its  two  points  of 
intersection  with  C.  If  C  and  F  have  a  point  P  in  common,  then 
to  P  on  C  correspond  two  coincident  tangents  to  F  at  P,  so  that 
their  corresponding  points  also  coincide  at  P.  This  is  only  pos- 
sible when  C  and  P  are  tangent  at  P.  From  this  it  follows  that 
the  two  conies  C  and  F  are  doubly  tangent,  and  as  there  is  no  dis- 
tinction analytically,  we  may  say  that  in  case  of  no  real  intersec- 
tions the  conies  C  and  F  have  two  imaginary  tangencies. 

3.  From  (3)  it  is  seen  that  to  a  pencil 

ax-\-  by+c+  X(a'x+  by  +  c')=o 

corresponds  a  range;  and  the  vertex  of  the  pencil  corresponds  to 
the  line  of  the  range.  The  converse  (apply  (4))  is  also  true.  Let 
now  U  and  5  be  the  points  of  tangency  of  C  and  F,  and  u  and  5 
the  tangents  at  U  and  S,  and  T  their  point  of  intersection,  and 
consider  the  planes  of  (u,  v)  and  (x,  y)  as  made  up  of  Hnes  and 
points  and  points  and  lines  respectively;  i.e.,  to  a  couple  (u,  v), 


THEORY  OF  CONICS.  129 

{au-\rhv-\-\^d)  in  one  plane  corresponds  dualistically  a  couple 
{x,  y),  {ax-\r^y+-L=o)  in  the  other  plane,  and  conversely. 

No  matter  whether  we  consider  T  as  belonging  to  one  or  the 
other  plane,  SU  is  the  corresponding  line  in  both  cases.  In  the 
reciprocal  transformation  T  and  SU  are  therefore  in  the  relation 
of  involution.     For  both  conies  5£/  is  the  polar  of  T. 

The  question  is  whether  it  is  possible  to  find  a  reciprocal  trans- 
formation for  which  the  involutoric  property  is  true  in  general. 
For  this  purpose  consider  a  point 

au-\-hv-\-c  =  o 

in  the  plane  {u,  v)  and  the  same  point  (  — ),  (  ~ )  in  the  plane  {x,  y). 

(a    h\  . 
To  the  point  (  — ,  —  I  in  the  x;y-plane  corresponds  the  line  with 

the  coordinates 

(7)  ^  = ,  77    , ,    v  = ,  77    , —  m  the  wz;-plane. 

To  the  point  {au+bv-{-c  =  o)  in  the  wiz-plane  corresponds  the 
line 

(aa^+ba2+  ca3)x-{-  (a6i  +  ^62+^^3)>'+  {aCi-^- bc^^ cc^)  =0 

in  the  xy-plsme.     Its  line-coordinates  are 

aa^+ba^-i-ca^  ab^+bb^+cbs 

(8)  U'  = -T , ,       V  =  —r . . 

For  an  involutoric  relation  the  two  lines  (7)  and  (8)  must  be 
identical.  This  will  be  the  case  when  b^^a^,  c^  =  (i3,  ^2  =  ^3;  i.e., 
if  the  transformation  (i)  has  the  form 

aiX+b^y+Ci  b^x+b^y+c^ 

(9)  ^  = \ \ — >     ^  = \ \ — • 

According  to  (2),  §  37,  these  are  the  formulas  for  a  transforma- 


13° 


PROJECTIVE  GEOMETRY. 


tion  by  reciprocal  polars.     To  prove  this  directly  the  equation  (5) 
of  the  conic  C  now  becomes 


(10) 


aiX^-{-  2hiXy-\-  h^y'^^  2CiX-\-  2C2y+  ^3=0. 


To  a  point  uXi+vyj^+i=o  now  corresponds  the  line  (accord- 
ing to  (4)) 

(11)     (aiXi+b^yi+  Ci)x+  (b^x^i-  b^y^^  C2)y+  (CiXi+C2yi+  c^)  =0. 

This,  however,  is  the  polar  of  the  point  {x^,  y^)  with  respect  to 
(10). 

An  involutoric  recipcocal  transformalion  is  therefore  a  trans- 
formation by  reciprocal  polars. 

In  this  case  the  conies  C  and  T  coincide. 

4.  The  line-coordinates  given  in  (7)  and  (8)  are  also  identical 
if  corresponding  numerators  and  denominators  are  proportional. 
Designating  the  proportionality  factor  by  A,  these  conditions 
assume  the  form 


(12) 


a^{i  —  X)a-{-{b^—Xa2)b  -\-{c^—Xa^c=o, 
{a2—Xb^a-\-  b2{T-  —  X)b+{c2—Xb^c=o, 
(a^— Xcj)a+ (b^— Xc2)b  +Cs{i  —  X)c  =0; 


but  consistency  of  these  equations  requires  the  vanishing  of  their 
determinant,  or 


(13) 


ai(i  —  X)  b^—Xa2  c^—Xa^ 
a^-Xb^  b2(i~X)  C2—Xb^ 
a^—Xci       bs—Xc2       Cs(i-X) 


This  is  the  case,  first  when  b^  =  a2,  c.^  =  a^,  C2  =  b^,  and  X=i,  as 
discussed  under  3;  secondly,  by  expanding  the  determinant 
according  to  ascending  powers  of  X  and  solving  the  cubic  in  X. 
Thus  three  values  for  X  are  obtained  which  make  the  determinant 
vanish.  One  of  these  values  is  always  real,  so  that  there  is  at 
least  one  real  line  which  with  its  corresponding  point  forms  an 
involutoric  couple. 


THEORY   OF  CONICS.  131 

To  push  the  investigation  of  involutoric  reciprocity  one  step 
further  we  may  put  the  condition  for  the  equahty  of  u  and  u\ 
and  V  and  v',  in  (7)  and  (8)  in  the  form 

(14)  {a^x-\- a^y+  a3){a^x+h^y+  c^)-  {a^x+\y+c^){c^x-]rC^y+c^)=o, 

(15)  (&iX+  &2>'+  ^3) («3^+  hy+c^) -  («2^+  hy-^-  C2) {c^x-^r  c^y+  Ca)  =  o, 

where  x=—,  y=^—  are  the  coordinates  of  the  original  point. 

There  are  generally  four  solutions  of  (x,  y)  which .  satisfy 
(14)  and  (15)  simultaneously.  Of  these,  one  is  the  point  of  inter- 
section of  the  lines  a^x-\-'b^y-\-c^  =  o  and  c^x-\-c^y-\-c^  =  o,  which, 
however,  is  to  be  excluded.  In  fact,  according  as  this  point  is 
considered  as  belonging  to  one  or  the  other  plane,  {u,  v)  or  (m',  ^''), 

u     aj^x-j-b^y-^-c^ 

the   lines  through   the   origin  with   the  slopes  —  = -r — 7~} 

^  ^  ^      V     a^x+h^y+c^ 

u'    a^x+a^y+as  a   ,     ;      n  i.         f       a 

—=-, j 7-   correspond   to   it.     Hence,   as  we   have   lound 

v'     b^x+b^y+b^ 

before,   there  are  in  general  only  three  involutoric  pairs  in  a 

reciprocal   transformation;     they   are   determined   by   the    three 

remaining  points  of  intersection  of   (14)   and   (15)  and  form  a 

triangle  UST,  according  to  3,  in  which  TU  and  TS  correspond 

to  the  points  U  and  S,  and  T  to  the  line  US,  involutorically. 

Hence   in  a   reciprocal    transformation  there  is  generally  only 

one  involutoric  pair  (T,  US)  which  is  not  coincident. 

Suppose  that  this  be  true  for  a  second  pair  of  this  kind,  then 
(14)  and  (15)  would  have  a  fifth  common  solution  which  is  only 
possible  when  the  two  are  identical.     Hence  the  theorem: 

//  a  reciprocal  transformation  contains  two  non-coincident 
involutoric  pairs,  then  all  its  pairs  are  involutoric;  the  transforma- 
tion is  a  so-called  polar  reciprocity. 

By  two  non- coincident  involutoric  pairs  the  polar  reciprocity  is 
fully  determined. 

To  prove  this  last  theorem  equations  (9)  and  the  equation  for 

u 

—  obtained  from  them  may  be  written  in  the  form 


132  PROJECTIVE  GEOMETRY, 

a.  h,  C.  ^      C2 

—  x-[-  —  y-{-—{i  —  ux) uy—u=o, 


b,        K       c,         C2  . 

—  xi-  —  y vx+—{i  —  vy)  —  v=o, 


a,         b,  &2         ^1       ^2 

—vx-^—^vy—  ux)  —  —uy+  —v—  — u=o. 


Giving  {x,  y)  two  arbitrary  values  and  (u,  v)  correspondingly 
two  arbitrary  values,  six  equations  with  the  only  five  unknown 

.    .  «!      &1        &2       ^1       ^2  1         •  1  T^       • 

quantities  — ,  — ,  — ,  — ,  — ,  are  obtained.     Designating  the  two 
^3    Q    ^3    ^3    ^3 

pairs  by  {x^,  y^),  (X2,  yz)  and  (u^,  v^),  {u^,  v^),  the  determinant  of 
the  six  equations  becomes 


^1 

X, 

Jl 

{l-U,X,) 

0 

-UJ, 

—  Ml 

-■^2 

X2 

yt 

{1-U2X2) 

0 

-^2^2 

-U2 

—  Ui 

0 

X, 

-v,x. 

yi 

(l-^l}'l) 

-■z^i 

^2 

0 

X2 

-^2^2 

y2 

(^-'^2y2) 

-"^2 

+  1 

v^x^ 

(v 

!>'] 

—  U^Xi) 

^1 

-u^y. 

—  % 

0 

—  I 

V^2 

{V 

2y 

I 

U2X2) 

^2 

-^2y2 

—  U2 

0 

In  fact  multiplying  the  six  rows  successively  by  v^,  —V2,  —u^,  Wj, 
+  1,  —I,  as  indicated,  after  this  multiplication,  the  sum  of  the 
first  four  rows  is  equal  to  the  sum  of  the  last  two,  which  shows 
that  any  of  the  six  equations  may  be  expressed  in  terms  of  the 
five  remaining  ones.  The  above  five  quantities  are  therefore 
uniquely  determined,  which  proves  the  theorem. 

5.  In  a  polar  reciprocity,  or  simply  in  a  polar  system,  two 
pairs  A^  a  and  P,  p  determine  at  once  a  third.  Indeed  the  fine 
c  joining  A  and  P  is  the  polar  of  the  point  of  intersection  C  of  a 
and  p.  The  pole  of  ^C  is  the  intersection  of  a  and  c,  say  B. 
Thus,  starting  with  two  pairs,  we  have  constructed  a  triangle 
ABC,  whose  vertices  are  the  poles  of  its  opposite  sides.  Such 
a  triangle  is  called  a  self- polar  triangle  (§  14).  Clearly  in  every 
polar  system  there  are  an  infinite  number  of  self- polar  triangles; 
but  by  such  a  triangle  a  polar  system  is  not  completely  determined. 


THEORY  OF   CONICS.  1 33 

To  do  this,  another  pair,  like  P,  p,  must  be  added  to  the  given 
triangle. 

6.  Without  following  the  subject  of  polar  systems  further  we 
remark  that  the  great  geometer  von  Staudt  has  made  it  the  back- 
bone of  his  geometry  of  position.  In  this  connection  conies 
appear  as  special  properties  of  polar  systems  and  no  distinction, 
or  separate  treatment  of  real  and  imaginary  elements,  is  necessary. 

In  view  of  the  various  methods  applied  in  this  work,  we  have 
found  it  advisable  to  be  satisfied  with  the  foregoing  short  account. 

It  would  be  very  valuable  if  some  geometer  could  show  how, 
with  polarity  as  a  base,  projective  geometry  might  be  made  as 
simple  and  as  accessible  to  the  appKcations  as  the  traditional 
methods. 

§  39.    Theorems  of  Pascal  and  Brianchon.^ 

I.  Assume  six  points  A,  B,  C,  D,  E,  F  in  any  order  on  a 
conic  and  consider  any  two  of  them,  say  A  and  C,  as  vertices 
of  pencils  of  rays  in  the  conic.  Fig.  49.     Then 

{A  ■  BCDEF)  =  C  ■  BCDEF). 

Cutting  these  pencils  by  the  Hnes  ED  and  EF  respectively, 
two  projective  point-ranges, 

(B,C,D,E,F,)  =  (B,C,D,E,F,), 

are  obtained,  and  as  E^  is  identical  with  Ej  it  follows  (§9)  that 
the  two  ranges  are  perspective.  Hence  B^B^,  C^Co,,,  D^D^,  .EiEj, 
^1^3  are  concurrent  at  a  point  ^3.  This  will  be  true  no  matter 
how  the  six  points  may  be  distributed  over  the  conic,  pro- 
vided the  foregoing  order  of  the  points  is  followed.  The  lines 
followed  in  the  order  ABC  DEE  form  now  a  closed  hexagon, 

^  Pascai  (1623-1662)  discovered  his  theorem  when  sixteen  years  of  age  and 
called  it  Hexagramma  Mysticum.  It  appeared  first  in  Pascal's  "Conic  Sections," 
which  was  published  in  1640. 

Brianchon  (1785-1864)  published  his  theorem  in  1806  in  the  Journal  de 
I'Ecole  Polytechnique,  Vol.  XIII. 


134 


PROJECTIVE  GEOMETRY. 


and  we  may  call  opposite  sides  of  this  hexagon  lines  which  are 
separated  by  two  adjacent  sides  of  the  hexagon,  which  is  all  in 
analogy  with  the  regular  hexagon.  The  pairs  of  opposite  sides 
intersect  at  B^,  B^,  B^,  three  collinear  points.     As  the  six  points 


Fig.  49. 

were  arbitrarily  selected,  this  is  generally  true,  hence  Pascal's 
Theorem: 

In  any  hexagon  which  is  inscribed  in  a  conic,  the  three  pairs 
of  opposite  sides  intersect  in  three  collinear  points. 

We  shall  call  such  a  line  of  collinearity  a  Pascal  line. 

By  reciprocation.  Fig.  50  (§  37),  we  obtain  immediately  in 
its  generaUty  Brianchon's  Theorem: 


Fig.  50. 

In  any  hexagon  which  is  circumscribed  about  a  conic,  the 
three  principal  diagonals  are  concurrent. 


THEORY  OF  CONICS.  I3S 

We  shall  call  such  a  point  of.  concurrence  a  Brianchon  point. 

2.  Salmon,  in  his  treatise  on  Conic  Sections,  1848,  gave  a 
remarkably  simple  proof  for  Pascal's  theorem,  based  upon  the 
abbreviated  designation  of  straight  lines  in  analytic  geometry. 
Let  A=o,  B=o,  C=o,  D=o,  £=0,  F=o  be  the  equations  of 
the  sides  of  any  hexagon  inscribed  to  a  conic,  and  G  =  o  the  equa- 
tion of  the  straight  line  joining  the  vertices  (^=0,  F  =  o)  and 
(C  =  o,  Z)=o).     Then 

A-C-XB-G=o,     F-D-[xE-G  =  o 

are  two  forms  in  which  the  equation  of  the  given  conic  may  be 
written.     From  these  two  forms  we  get 

A-C-F-D^G{XB-iiE). 

Now  the  points  (^=0,  D=o)  and  (C=o,  F  =  o)  are  not 
situated  on  the  Hne  G=o,  consequently  they  must  He  on  the 
line  XB— ^E  =  o.  In  other  words,  the  points  (-4  =0,  D=o), 
(C  =  o,  F  =  o),  (B=o,  E  =  o)  are  collinear,  and  as  they  are  the 
points  of  intersection  of  pairs  of  opposite  sides  in  the  hexagon, 
Pascal's  theorem  is  proved. 


Fig.  51. 


By  considering  A=o,  etc.,  as  the  line-equations  of  the  six 
vertices  of  a  hexagon  circumscribed  to  a  conic,  Brianchon's 
theorem  may  be  deduced  in  a  similar  manner. 


136  PROJECTIVE  GEOMETRY. 

3.  Assuming  as  a  conic  a  degenerate  hyperbola,  consisting 
of  two  intersecting  lines  and  on  each  three  points,  say  A,  E,  C 
and  D,  B,  F,  Pascal's  theorem  still  holds;  i.e.,  the  points  B^, 
B2,  Bs  are  collinear. 

If  AB  \\DE  and  EF  ||  CB,  then  5^  and  B2  and  consequently 
also  Bs  are  infinitely  distant;  i.e.,  also  CD  \\AF,  Fig.  51.  Hence 
the  special  theorem: 

//  on  each  of  two  intersecting  lines  three  points  A,  C,  E  and 
B,  D,  F  are  chosen,  so  that  AB  is  parallel  to  DE  and  EF  parallel 
to  BC,  then  CD  is  also  parallel  to  AF. 

In  this  special  form  Hilbert  in  his  Foundations  of  Geometry,^ 
p.  28,  uses  Pascal's  theorem  to  establish  a  non-Archimedean 
geometry. 

Ex.  I.  Prove  Pascal's  special  theorem  directly. 

Ex.  2.  EstabHsh  the  dualistic  of  Ex.  i. 

Ex.  3.  li  A=o,  rA  +  bB  =  o,  yB'  +  aA'^o,  A'  =  o,  aA'+fB 
=  0,  fA-\-hB'  =  o  (where  a,  h,  y,  f  are  numerical  factors  and 
A,  B,  A',  B'  hnear  expressions  in  x  and  y)  are  the  sides  of  a  hex- 
agon, prove  that  this  hexagon  is  inscribed  to  the  conic 

aAA'-hBB'  =  o, 
and  that 

yfA  -  abA'  =  o 

is  the  Pascal  line  of  the  hexagon.  (BobilHer,  1828.) 
'  Ex.  4.  If  six  points  on  a  conic  are  given,  it  is  possible  to  pass 
in  five  different  ways  from  any  point  to  the  others.  From  each 
of  these  four  different  paths,  not  chosen  before,  may  be  taken  to 
join  the  remaining  points;  from  each  of  these  three  different 
paths  may  be  selected ;  and  so  forth.  Finahy  the  original  point 
is  reached  in  5-4-3-2-i  =  i2o  ways;  but  as  each  closing  side  is 
contained  in  one  of  the  original  paths,  it  is  evident  that  only 
120:2  =  60  different  closed  hexagons  can  be  formed.  Hence 
with  six  points  on  a  conic  may  he  formed  sixty  different  hexagons 
and  consequently  sixty  different  Pascal  lines. 

^  Griindlagen  dsr  Gcometrie,  Teubner,  Leipzig,  1899. 


THEORY  OF  CONICS.  137 

Between  these  lines  exist  a  number  of  interesting  relations.* 

Verify  the  following  propositions  in  a  regular  hexagon: 

The  sixty  Pascal  lines  intersect  each  other  three  by  three  in 
twenty  points  G  (Steinerian  points).     (Steiner's  theorem.) 

Besides  these  points  G,  the  sixty  Pascal  lines  have,  three  by 
three,  sixty  other  points  H  in  common.     (Kirkmann's  theorem.) 

There  are  twenty  lines  g  each  of  which  contains  a  point  G 
and  two  points  H.  Four  by  four  of  these  lines  pass  through 
fifteen  points  J.     (Cayley's  theorem.) 

The  points  G  lie  four  by  four  in  fifteen  straight  lines  J. 
(Steiner's  theorem.) 

Designating  the  original  six  points  by  123456,  then  a  Steiner- 
ian point  is  given  by  the  intersection  of  the  Pascal  lines  of  the 
three  hexagons  123456,  143652,  163254. 

Ex.  5.  State  the  dualistic  of  the  foregoing  theorems. 


§  40.   Applications  of  Pascal's  and  Brianchon's  Theorems. 

1.  Construction  of  a  conic  when  five  of  its  points  are  given. 

The  practical  importance  of  Pascal's  and  Brianchon's  theorem 
lies  in  the  possibility  of  constructing  an  unlimited  number  of 
points  and  tangents  of  a  conic,  when  five  of  its  determining  ele- 
ments are  given. 

Let  ABODE  be  five  points  of  a  conic  and  AB,  BC,  CD,  DE 
four  consecutive  sides  of  an  inscribed  hexagon.  In  Fig.  52,  it 
is  clear  that  the  Pascal  line  p  passes  through  B^,  the  point  of  in- 
tersection oi  AB  and  DE.  Now  there  are  an  infinite  number  of 
points  F  possible  on  the  conic  and  consequently  an  infinite  num- 
ber of  Pascal  lines  through  5i.  Thus  to  every  point  F  on  the 
conic  corresponds  one  Pascal  line  through  B^.  Hence,  assuming 
any  line  p  through  Bj^,  the  line  EF  passes  through  the  intersection 
Bs  of  BC  and  p.  In  a  similar  manner  the  fine  FA  is  obtained 
by  joining  A   with  the  point  of  intersection  B^  of  CD  with  p. 

^  See  Salmon-Fiedler,  Analytische  Geometrie  der  Kegelschnitte,  Vol.  II,  pp. 
459-466,  5th  edition. 


138 


PROJECTIVE  GEOMETRY. 


The  point  where  the  produced  lines  of  EB^  and  AB^^  meet  is 
evidently  the  required  point  F  on.  the  conic,  corresponding  to  the 


Fig.  52. 

chosen  Pascal  line  p.     Repeating  the  same  construction  for  every 
line  p  through  B^,  all  points  of  the  conic  are  obtained. 

To  construct  the  tangent  at  any  point  of  the  conic,  say  A, 
consider  F  infinitely  close  to  A.  Apply  the  general  construction 
of  p  for  ABC  DBF,  then  the  line  joining  B2  with  A  is  the  tangent 
at  this  point. 

2.  Construction  of  a  conic  when  five  of  its  tangents  are  given. 

Let  a,  b,  c,  d,  e  be  the  given  tangents.  Fig.  53,  forming  five  con- 
secutive sides  of  a  circumscribed  hexagon.  The  Hne  b^  joining 
the  points  of  intersection  of  a  and  b,  and  d  and  e,  passes  through 
the  Brianchon  point  P.  Now,  every  sixth  tangent  determines 
another  point  P  on  b^.  Conversely,  every  point  P  on  ^^  deter- 
mines a  sixth  tangent  of  the  conic.  Thus,  to  find  a  sixth  tangent 
/,  assume  any  point  on  b^  as  the  Brianchon  point  P.  Then  the 
line  through  be  and  P  will  be  the  Hne  b^  cutting  e  in  the  point 
where  also  /  cuts.  In  a  similar  manner,  the  line  joining  the  point 
of  intersection  of  c  and  d  with  P  is  b^,  which,  when  produced,  cuts 
a  in  the  same  point  as  /.  Hence  the  line  joining  the  points  of 
intersection  of  e  and  &2>  and  63  and  a,  is  the  sixth  tangent  corre- 
sponding to  the  chosen  P.  Repeating  this  construction  for  all 
points  of  &i,  all  tangents  of  the  conic  are  obtained. 


THEORY   OF   CONICS.  139 

To  construct  the  point  of  tangency  of  any  tangent,  we  may 
consider  this  one  as  two  coincident  tangents  (consecutive),  say  a 
and  /,  and  these  with  the  remaining  four,  when  subject  to  the 
general  construction  of  the  Brianchon  point,  lead  to  the  required 
point  of  tangency. 

3.  By  the  same  methods  conies  may  also  be  constructed  when 
they  are  determined  by  mixed  elements;  i,e.,  points  and  tangents, 
always  five  in  number.     In  these  problems  a  tangent  appears  as 


Fig.  53. 

a  line  joining  two  consecutive  points,  and  a  point  as  the  point  of 
intersection  of  two  consecutive  tangents.  The  same  construc- 
tions may  also  be  extended  to  cases  where  one  point,  two  points, 
or  one  tangent  is  infinitely  distant. 

Ex.  I.  Given  five  points  of  a  conic;  to  construct  the  tangents 
at  these  points. 

Ex.  2.  The  dualistic  of  Ex.  i, 

Ex.  3.  Given  three  points  and  the  directions  of  the  asymp- 
totes of  an  hyperbola;  to  construct  any  number  of  points  of  the 
hyperbola. 

Ex.  4.  Given  four  tangents  of  a  parabola  (one  tangent  is 
infinitely  distant).     To   construct   any  number  of  its   points. 

Ex.  5.  Given  four  points  and  a  tangent  of  a  conic;  construct 
other  points  of  the  conic. 


I40  PROJECTIVE   GEOMETRY. 

Ex.  6.  Dualistic  of  Ex.  5. 

Ex.  7.  Given  three  points  and  two  tangents  of  a  conic.  To 
construct  it.     Also  make  the  duahstic  construction. 

Ex.  8.  Given  three  points,  a  tangent,  and  its  point  of  tan- 
gency;  construct  the  conic. 

Ex.  9.  Given  the  two  tangents  at  the  given  vertices  of  an 
eUipse  or  hyperbola  and  a  third  tangent;  to  construct  any 
number  of  tangents. 

Ex.  10.  Given  the  two  asymptotes  and  a  tangent  of  an  hyper- 
bola; to  construct  it. 

Ex.  II.  Given  the  axis,  vertex,  and  two  other  points  of  a 
parabola;    construct  it. 

Ex.  12.  Given  three  points  and  an  asymptote  of  an  hyper- 
bola;  to  construct  it. 


§  41.   Conies  in  Mechanical  Drawing  and  Perspective. 

I.  To  inscribe  an  ellipse  in  a  parallelogram. 

The  middle  points  of  the  sides  shall  be  the  points  of  tangency 
of  the  ellipse.  Two  points  of  tangency  may  be  designated  hy  AB 
and  CD,  and  the  third  by  E,  Fig.  54.  The  explanation  of  the  con- 
struction of  points  of  the  ellipse  by  Pascal's  theorem  is  identical 
with  that  of  Fig.  52,  §  40,  and  is  apparent  from  Fig.  54.  By 
assuming  a  second  Pascal  line  through  L  with  points  H  and  / 
corresponding  to  M  and  N  on  the  first  Pascal  Hne,  a  second 
point  G  is  obtained.  The  same  construction  repeated  for  other 
Pascal  lines  through  L  gives  further  points  of  the  ellipse,  so 
that  the  ellipse  through  these  points  may  be  sketched  free-hand 
or  by  mans  of  a  curved  ruler.  In  this  figure  the  ellipse  appears 
manifestly  as  the  product  of  two  projective  pencils  with  A  and 
E  as  vertices.     In  fact, 

{AMCH)  =  {KNCJ), 

since   these  points  are  projected  by   one   and   the   same   pencil 
through  L.     Taking   a   Pascal  line  parallel   to   KC   and  desig- 


THEORY    OF  CONICS 


141 


nating  its  point  of  intersection  with  AC  produced  by  7,  then 
the  point  corresponding  to  /  has  moved  to  infinity  on  KC^  and 


Fig.  54. 

to  the  Pascal  Hne  LI  corresponds  the  point  E  on  the  ellipse. 
Now 

(AMCI)  =  {KNC^), 

and  as  AC  =  CIi  these  ratios  become 

I  AM     KN 
2'CM~  CN' 

But  there  is  also 


{AMCI)  =  (PF^C^)\ 


hence 


KN    OF, 

CN~CF[' 


142 


PROJECTIVE   GEOMETRY. 


From  this  it  follows  that  the  rays  AN  and  EM  of  the  pencils 
through  A  and  E  trace  on  KC  and  OC  similar  point-ranges. 
If,  therefore,  KC  and  OC  are  divided  into  any  number  of  equal 
parts  and, the  division-points  are  numbered  from  K  io  C  and 
from  O  to  C,  then  the  rays  joining  E  and  A  with  equal  numbers 
on  KC  and  OC  intersect  each  other  in  points  of  the  ellipse.  In 
a  similar  way  this  construction  may  be  extended  to  the  remain- 
ing three  quarters  of  the  ellipse.  The  same  method  may  obviously 
be  apphed  to  rectangles  and  squares.     See  Figs.  37,  38;  §  27. 

2.  To  inscribe  an  ellipse  to  any  quadrilateral. 

A  quadrilateral  may  be  considered  as  the  perspective  of  a 
square,  and  it  must  therefore  be  possible  to  apply  the  previous 
construction  to  any  quadrilateral.  The  distances  KC  and  OC, 
Fig.  55,  must  now  be  divided  perspectively  into  a  number  of 
equal  parts.  The  fundamental  principle  of  perspective  division 
is  the  following: 

//  KC  as  a  side  of  a  rectangle  AKCO,  in  perspective,  shall  be 
divided  into  two  equal  parts,  draw  the  diagonals  AC  and  KO  and 
join  their  point  of  intersection  W  to  the  point  X,  where  A K  and.  OC 
produced  meet.     WX  cuts  KC  at  its  middle  point,  M. 


Fig.  55. 


In  the  first  place,  the  points  AB,  C,  E,  etc.,  were  obtained  by 
the  application  of  this  principle  to  the  given  quadrilateral. 

By  the  same  principle  KM  and  CM  may  be  again  bisected. 


THEORY   OF  CONICS.   ■  143 

OC  may  be  subdivided  in  the  same  manner.  Fig.  38,  §  27,  illus- 
trates the  construction  of  an  elhpse  inscribed  in  a  quadrilateral 
by  this  principle. 

The  problem  to  inscribe  an  ellipse  into  a  quadrilateral  appears 
in  a  great  number  of  special  forms  in  perspective.  For  example, 
a  trapezoid  may  be  considered  as  the  perspective  of  a  square 
having  two  opposite  sides  parallel  to  the  picture-plane,  as  in 
the  case  of  window- frames  and  doors. 

3.  To  construct  a  parabola  Jiaving  the  vertex,  the  major  axis, 
and  a  point  given. 

Let,  in  Fig.  56,  the  vertex  be  designated  by  AB,  the  infinitely 
distant  point  of  the  axis  by  DE,  and  the  third  point  by  C.  Evi- 
dently any  line  p  parallel  to  the  tangent  at  the  vertex  may  be 
considered  as  a  Pascal  line.     The  construction 

AB  I         BCl  CD  I 

DE  S    '     EF\      '     FA   ) 

for  the  assumed  Pascal  hne  p  gives  us  a  point  F  of  the  parabola. 
If  p  varies,  the  point-ranges  traced  by  M,  N',  F.^  on  AC,  KC,  AK 
are  all  similar.  Hence,  dividing  KC  and  AK  in  any  number 
of  equal  parts,  numbering  the  division-points  from  K  to  C  and 
from  A  to  K,  a  line  joining  A  to  any  number  on  KC  and  a  line 
through  the  equal  number  on  AK,  parallel  to  the  axis,  cut  each 
other  in  a  point  of  the  parabola. 

4,  Construction  0}  a  parabola  which  is  the  junicular  polygon  of 
a  uniformly  distributed  load  on  a  horizontal  beam. 

If  a  load  is  uniformly  distributed  on  a  horizontal  beam,  then 
the  funicular  polygon  is  a  parabola  hmited  by  points  in  perpen- 
diculars through  the  extremities  of  the  beam.  The  tangents  at 
the  extremities  of  the  parabola  are  known;  they  are  parallel  to 
the  extreme  lines  of  the  force  polygon.  Designate  in  Fig.  S7 
the  tangents  at  the  extremities  by  ab  and  de,  and  the  infinite 
tangent  by  c.  Then  the  Brianchon  points  P  are  situated  on  ad, 
and  hues  through  P  parallel  to  ab  and  de  (the  tangents  at  the 
extremities)  cut  these  in  two  points  x  and  y  through  which  the 


144 


PROJECTIVE   GEOMETRY. 


sixth  tangent  /  passes.     If  P  moves  on  ad,  then  the  points  x  and  y 
trace  onab-Q  and  de  ■  Q  two  equal  point- ranges.     Hence,  dividing 


Fig.  56. 

these  distances  in  a  number  of  equal  parts  and  numbering  them 
from  ab  and  Q,  the  lines  joining  equal  numbers  are  tangents  of 
the  required  parabola.  To  find  the  point  of  tangency  of  /, 
replace  ab  by  c,  de  by  /,  /  by  ab,  and  the  infinite  tangent  by  de, 


de  being  the  infinite  point  of  the  axis  of  the  parabola.  For  this 
arrangement  P  is  also  the  Brianchon  point,  and  the  construction 
shows  at  once  that  the  required  point  of  tangency  T  is  cut  out  by 
a  line  through  P  parallel  to  the  axis  of  the  parabola. 


THEORY   OF  CONICS. 


U5 


To  sum  up,  we  have  the  following  construction  for  a  parabola 
touching  the  sides  of  an  isosceles  triangle  ABC,  AC  =  BC,  at  A 
and  B:  Divide  AB,  BC,  and  CA  into  the  same  number  of  equal 
pans  and  number  the  division-points  from  A  to  B,  jrom  A  to  C, 
and  jrom  C  to  B.  The  lines  joining  equal  numbers  on  AC  and 
CB  are  tangents  of  the  required  parabola,  and  the  perpendiculars 
from  corresponding  equal  division- points  on  AB  cut  these  tangents 
in  iheir  points  of  tangency. 

5.  Construction  of  an  equilateral  hyperbola  when  its  asymptotes 
and  the  tangent  at  a  vertex  are  given. 

In  Fig.  58  designate  the  asymptotes  by  ab  and  de,  and  the 
tangent  at  the  vertex  by  c.     Let  c  cut  the  asymptotes  at  A  and  B. 


>a'a6 


In  this  case  the  Brianchon  points  are  infinitely  distant.  Hence, 
drawing  through  A  and  B  two  parallel  lines  in  any  direction, 
cutting  the  asymptotes  at  C  and  D,  the  line  joining  C  with  D 
will  be  a  required  tangent  of  the  hyperbola.  To  study  the  metrical 
relations  of  this  hyperbola,  we  have 

AAODc^ACOB, 


146  PROJECTIVE   GEOMETRY. 

hence  DO -.AO^ BO -.CO, 

or  CO  ■  DO  =A0- BO  =  consta.nt. 

Designating  the  distance  of  the  vertex  M  from  the  asymptotes 
by  k,  there  evidently  is 

C0D0  =  4k^ 

a  relation  which  holds  for  any  tangent.  Hence  the  triangle 
between  the  asymptotes  and  any  tangent  has  a  constant  area.  (This 
is  true  for  any  hyperbola,  as  might  be  proved  in  a  similar  manner.) 

To  find  the  point  of  tangency  of  CD,  replace  the  designation 
7  by  ab,  ah  by  de,  de  by  c,  and  AB  by  /.  (The  student  should 
make  a  new  figure.)  Join  the  point  of  intersection  Q  oi  AB 
and  CD  to  O  and  produce  to  the  point  of  intersection  R  with 
BC;  then  R  is  the  new  Brianchon  point  and  the  line  through 
R  parallel  to  OD  cuts  CD  in  the  required  point  of  tangency  N. 
As  A  BCD  is  a  quadrilateral  in  which  AD  \\  BC  and  O  and  Q 
are  diagonal  points,  BR  =  RC,  hence  also  DN  =  CN.  The  point 
of  tangency  bisects,  there] ore,  the  tangent  between  the  asymptotes 
{general  proposition). 

Designating  the  coordinates  of  N  by  x  and  y,  we  have  x=^CO, 
y  =  \DO;  hence  xy  =  \CO-DO,  and  as  CO-DO  =  /^k\ 

xy  =  k'^. 

This  is  the  equation  of  the  hyperbola  referred  to  its  asymptotes. 
A  full  treatment  of  this  case  was  given  in  view  of  its  importance 
in  the   graphical  representation   of  Boyle's   law  expressing  the 
relation  of  the  volume  x  and  the  pressure  ;y  of  a  gas. 

§  42.   Special    Constructions    of    Conies    by    Central    Projection 
and  Parallel  Projection.^ 

I.  Given  five  points  oj  a  conic,  to  constnict  a  circle  of  which  the 
given  conic  is  a  perspective. 

^  For  the  collection  of  these  problems  the  author  is  indebted  to  Dr.  Karl  Doehle- 
mann's  Geometiische  Transformationen,  I.  Teil,  Goschen,  Leipzig,  1902. 


THEORY    OF   CONICS.  147 

In  §§  20  and  27  it  has  been  shown  analytically  and  synthetic- 
ally that  evety  quadrilateral  may  be  considered  as  the  perspec- 
tive of  a  rectangle  which  is  always  inscribed  to  a  certain  circle. 
It  is  therefore  possible  to  construct  four  points  A,  B^  C,  D  oi  a, 
rectangle  as  the  points  whose  perspectives  are  four  given  points 
A',  B',  C,  D'  of  the  conic.  As  has  been  explained  in  §  27,  the 
two  diagonal  points  M'  and  N'  determine  the  vanishing-line  and 
are  the  vanishing-points  of  the  pairs  of  parallel  sides  AB,  CD  and 
AD,  BC.  The  center  of  perspective  colhneation  is  situated  on  a 
circle  over  M'N'  as  a  diameter,  and  AB,  CD  and  AD,  BC  are 
respectively  parallel  to  SM'  and  SN' ,  Fig.  59.     The  axis  of  col- 

q'  m'  p'  n'  q' 

1-^^.      %~^ TP^l 


^^B' 


Fig.  59. 

lineation  5  must  be  chosen  parallel  to  q'  or  M'N'  at  any  distance 
from  it.  From  5  and  s,  A  BCD  is  perfectly  determined.  To 
determine  its  position  in  space  the  distance-circle  with  5  as  a 
center  must  be  given.  There  are  therefore  three  elements,  S,  s, 
and  distance-circle,  which  determine  A  BCD,  of  which  A'B'C'D' 
is  a  perspective,  completely.  Hence  there  are  co^  rectangles  in 
space  of  which  A'B'C'D'  is  a  perspective.  If  now  E,  of  which 
H  is  the  perspective,  shall  also  be  situated  on  the  circle  through 


148  PROJECTIVE  GEOMETRY. 

A  BCD,  notice  that  ^C  is  a  diameter,  hence  A  EC  sl  right  angle. 
Consequently  if  we  produce  A'E'  and  C'E'  to  their  intersections 
P'  and  Q'  with  g',  the  center  5  necessarily  also  lies  on  the  circle 
over  P'Q'  as  a  diameter. 

In  the  figure  the  construction  of  ABCDE  has  been  removed 
parallel  to  ^  in  order  to  make  it  clearer.  In  this  construction  we 
may  dispose  arbitrarily  of  s  and  of  the  distance -circle.  Hence 
there  are  00  ^  circles  in  space  which  may  be  transformed  into  a 
given  conic  by  perspective,  under  the  given  conditions.  To  make 
this  proposition  general  it  must  be  remembered  that  the  analytic 
expression  for  perspective  involves  three  essential  parameters.  If 
a  translation  of  the  center  of  perspective  is  added,  two  more  con- 
ditions enter,  so  that,  together  with  the  choice  of  the  distance- 
circle,  six  constants  perfectly  determine  a  central  projection.  If, 
therefore,  the  general  equation 

ax^+2bxy  +  cy^+2dx+2ey-\-  f  =  o, 

by  means  of  this  projection,  is  transformed  into 

Ax^-{-2Bxy  +  Cy^+2Dx-}'2Ey  +  E  =  o, 

this  equation  contains  those  six  constants.  If  this  equation  shall 
represent  a  circle,  the  conditions  A='C,  B=o  must  be  satisfied, 
so  that  of  the  six  constants  only  four  remain  independent. 

Hence  the  theorem: 

A  conic  may  he  considered  as  central  projection  0/  00  *  circles  in 
space. 

Ex.  I.  Given  five  tangents  of  a  conic;  to  construct  a  circle  of 
which  the  given  conic  is  a  perspective. 

Hint:  Any  four  of  the  given  tangents  may  be  transformed 
into  a  rhomb  circumscribed  to  the  required  circle.  The  diagonals 
of  this  rhomb  are  perpendicular  and  intersect  at  the  center  M  of 
the  required  circle.  Furthermore,  the  piece  of  the  fifth  tangent 
between  two  parallel  sides  of  the  rhomb  appears  under  a  right 
angle  from  M. 

Ex.  2.  Any  two  conies  in  a  plane  may  he  considered  as  the  cen- 
tral projection  of  two  circles,     (^longe.) 


THEORY   OF  COXICS.  I49 

The  two  circles  are  supposed  to  be  in  one  and  the  same  plane. 
Now  every  point  of  the  plane  may  be  taken  as  the  center  of  pro- 
jection, so  that  there  are  oo^ -00^=00°  central  projections  of  a 
plane.  Transforming  the  equations  of  the  given  conies,  six 
parameters  are  introduced  of  which  we  may  dispose  arbitrarily. 
In  order  that  the  transformed  equations  represent  circles,  four 
conditions  must  be  satisfied,  so  that  there  is  still  an  infinite 
number  of  possibilities  for  the  problem  left. 

In  case  that  the  given  conies  have  four  real  points  of  inter- 
section, imaginary  elements  are  introduced  in  the  solution.  The 
validity  of  the  geometrical  problem  in  this  case  is  maintained  by 
Poncelefs  principle  of  continuity.''- 

Ex.  3.  Prove  that  any  conic  and  a  straight  line  in  its  plane 
may  be  projected  centrally  into  a  circle  and  the  infinite  line  of  its 
plane. 

2.  Conies  as  intersections  of  right  cones. 

Let  in  a  plane  perpendicular  to  the  paper,  Fig.  60,  a  conic  K 
with  the  foci  F,  F^  and  the  vertices  A,Ai  be  given.  At  one  of  the 
foci,  say  F,  construct  any  sphere,  S,  tangent  to  the  plane  of  the 
conic,  and  from  A  and  A  ^  draw,  in  the  plane  of  the  paper,  two  tan- 
gents to  this  sphere,  intersecting  at  V.  Consider  V  as  the  vertex 
of  a  cone  tangent  to  S;  this  cone  will  be  a  right  cone  cutting  the 
plane  of  iT  in  a  certain  conic  iv'  with  the  same  vertices  A  and 
A^.  Let  the  cone  touch  the  sphere  along  the  circle  whose  plane 
is  T  and  which  cuts  the  plane  oi  K  in  a,  line  perpendicular  to  the 
plane  of  the  paper.  This  line  appears  as  a  point  D.  Assume 
any  point  P'  on  K'  and  let  P'V  cut  T  inQ;  then  P'Q  =  P'F  (in 
space).  The  true  length  of  P'F  is  P'R,  which  is  parallel  to  VA. 
Now,  no  matter  where  P'  is  taken  on  K' ,  P'R/FD  =  P'F/P'D^ 
constant.  This  constant  is  also  equal  AB/AD=AF  /AD. 
Hence,  as  P'  is  the  locus  of  the  points  whose  distances  from  a 
fixed  point  F  and  a  fixed  line  {D)  have  a  constant  ratio,  it  must 

^  Stated  by  Poncelet  in  the  introduction  of  his  Traite.  It  consists  in  the 
assumption  that  if  one  figure  is  obtained  from  another  figure  by  a  continuous 
variation,  then  projective  properties  derived  from  the  first  figure  also  hold  for  the 
second  figure.     The  principle,  however,  is  rigorous  only  when  proved  analytically. 


ISO 


PROJECTIVE  GEOMETRY. 


be  a  conic  with  the  focus  F  and  the  directrix  {D)  and  is  therefore 
identical  with  the  original  conic  K. 

In  Fig.    60  K  has  been    assumed  as  an  elhpse.     For  every 
sphere  tangent  at  F  there  is  consequently  a  right  cone  tangent 


Fig.  60. 
to  it  and  of  which  the  given  ellipse  is  a  section.     Now 
VA,-VA  =  VB,+B,A,-VB-BA=B,A,-BA=A^F-AF=FF^; 
i.e.,  VA 1—  VA  =  FF^  =  constant. 

V  moves,  therefore,  on  an  hyperbola  having  A^  A^  as  foci  and 
F,  Fi  as  vertices. 


THEORY   OF  CONICS.  151 

If,  in  place  of  an  ellipse,  an  hyperbola  is  chosen  for  K,  V  will 
be  on  an  ellipse  having  the  foci  of  the  hyperbola  as  vertices  and 
its  vertices  as  foci. 

To  sum  up  we  have  the  theorem: 

The  locus  0}  the  vertices  of  all  right  cones  passing  through 
a  given  ellipse  is  an  hyperbola  having  the  vertices  of  the  ellipse 
as  foci  and  it's  foci  as  vertices,  and  whose  plane  is  perpendicular 
to  the  plane  of  the  ellipse. 

The  locus  of  the  vertices  of  all  right  cones  passing  through  a 
given  hyperbola  is  an  ellipse  whose  vertices  and  foci  coincide  with 
the  foci  and  vertices  of  the  ellipse,  and  whose  planes  are  perpen- 
dicular to  each  other. 

Ex.  Prove  that  the  locus  of  the  vertices  of  all  right  cones 
passing  through  a  given  parabola  is  a  parabola  having  the  vertex 
of  the  first  as  a  focus  and  the  focus  as  a  vertex.  The  planes  of 
the  two  parabolas  are  perpendicular. 

That  there  are  no  other  right  cones  in  these  problems  with 
the  enumerated  properties  follows  from  the  fact  that  in  every 
right  cone  and  one  of  its  plane  sections  there  is  only  one  plane 
of  symmetry  with  respect  to  the  conic  section.  Conversely,  if  a 
conic  is  given,  the  vertex  of  a  right  cone  can  only  be  in  this  plane 
of  symmetry,  the  plane  passing  through  the  foci  and  perpendicular 
to  the  plane  of  the  conic. 

3.  Perspective  between  any  two  given  conies. 

Let  K  =  o  and  K' =  0  be  the  equations  of  any  two  conies  in 
the  same  plane.  Apply  a  general  perspective  collineation  to  K, 
thus  introducing  five  arbitrary  parameters  into  the  transformed 
equation.  In  order  to  make  this  last  equation  identical  with 
K' =  0,  corresponding  coefficients  must  be  set  equal.  This  gives 
five  equations  between  the  five  parameters  of  the  perspective  col- 
lineation, and  as  these  equations  are  of  the  second  degree  there 
will  be  several  solutions  of  the  problem.  Two  conies  in  a  plane 
may  therefore  always  be  considered  as  perspectives  of  one  another. 
Without  discussing  the  possibilities  of  real  and  imaginary  solu- 
tions of  these  equations  the  case  will  be  considered  where  K 
and  K'  have  .four  real  points  of  intersection  i,  2,  3,  4,  Fig.  61. 


152 


PROJECTIVE  GEOMETRY. 


K  and  K'  have  the  self-polar  triangle  XYZ  in  common.  Desig- 
nate the  four  common  tangents  by  I,  II,  III,  IV,  and  consider 
the  points  of  intersection  S  of  III  and  IV,  and  S^  of  I  and  II. 
Evidently  the  centers  of  perspective  must  be  sought  in  such 


Fig.  6i. 


points  of  intersection  of  common  tangents,  because  a  tangent 
from  the  center  of  perspective  to  one  conic  is  also  a  tangent  to 
the  perspective  conic.  The  common  chords  12  and  34  as  well 
as  the  chords  of  contact  A  B  and  A  'B'  pass  through  X  when  pro- 
duced. -  Choosing  34  as  the  axis  of  a  central  collineation,  5  as 


THEORY   OF  CONICS.  153 

the  center,  A,  A'  as  corresponding  points,  then  the  colhneation 
is  perfectly  determined,  and  the  conic  K  is  transformed  into  a 
conic  K",  which,  however,  is  identical  with  K' ,  since  it  has  the 
points  3,  4  and  the  points  of  tangency  A',  B' ;  i.e.,  six  points  in 
common  with  K' .  Instead  of  5  we  may  also  choose  the  chord  s^ 
as  the  axis,  and  5  as  the  center  of  a  colhneation.  Hence  with  5 
as  a  center  there  are  two  central  collineations  transforming  K 
into  K' .  Conversely,  every  chord,  as  s,  may  serve  for  two  colhnea- 
tions  with  5"  and  S^  as  centers.  The  same  can  be  analogously 
proved  for  every  common  chord  and  point  of  intersection  of  two 
common  tangents.     We  have  therefore  the  theorem : 

//'  iwo  conies  K  and  K'  have  jour  real  points  0}  intersection, 
then  there  are  12  central  collineations  in  which  K  and  K'  correspond 
to  each  other.  For  every  point  0}  intersection  of  two  common 
tangents  there  are  iwo  chords  which  may  be  taken  as  axes  of  iwo 
of  those  12  collineations.  Conversely,  to  every  chord  belong  two 
points  of  intersection  of  common  tangents  as  centers  of  two  such 
collineations. 

These  propositions  admit  of  an  easy  interpretation  in  space. 
As  every  common  chord  determines  two  centers  of  colhneation, 
it  follows  that  there  are  two  cones  through  iwo  conies  in  space  with 
two  points  in  common. 

In  §  T,T,  it  has  been  shown  that  on  account  of  the  rectangular 
polar  involution  around  the  center  of  colhneation  not  being 
changed,  a  circle  concentric  with  the  center  of  colhneation  is 
transformed  into  a  conic  whose  focus  is  in  this  center.  Gen- 
erally, for  the  same  reason,  a  conic  one  of  whose  foci  coincides 
with  the  center  of  colhneation  is  transformed  into  a  conic  having 
the  same  focus. 

But  this  is  also  in  agreement  with  the  previous  result.  A 
focus  of  a  conic  may  be  considered  as  the  point  of  intersection 
of  two  conjugate  imaginary  tangents  from  the  circular  points. 
Two  conies  with  the  same  focus  have  therefore  two  common 
imaginary  tangents,  and  their  real  point  of  intersection  may 
be  assumed  as  a  center'  of  colhneation  between  the  two  conies. 

Ex,  I.  Discuss    the  case   and    make    the  construction   when 


154 


PROJECTIVE  GEOMETRY. 


K  and  K'  intersect  in  two  real  points  and  have  two  parallel  tan- 
gents. 

Ex.  2.  Make  the  construction  when  K  and  K'  are  tangent. 

Ex.  3.  Discuss  the  arrangements  of  K  and  K'  in  order  to 
obtain  all  special  cases  of  perspective  coUineation. 

4.  Given  five  points  of  a  conic  K, — A,  B,  C,  D,  E;  through 
two  of  these,  say  A  and  B,  pass  a  circle  K' ,  and  find  the  center  of 
perspective  S  for  which  K  and  K'  are  corresponding. 

In  Fig.  62  construct  the  pole  X  oi  s  orAB,  which  we  assume 
as  the  axis  of  collineation.  This  is  easily  done  by  means  of 
the  points  of  intersection  P  and  Q  of  CD  and  DE  with  5  and 
their  polars  p  and  q  in  the  quadrilaterals  A  BCD  and  AEBD. 
Construct  also  the  pole  X'  of  5  with  respect  to  K',  then  X  and 
X^  are  corresponding  points  in  the  collineation  and  the  center  S 


Fig.  62. 

must  be  on  the  line  joining  X  with  X\  DX  and  D'X'  meet  on 
s,  in  a  point  R;  hence  X'R  cuts  K'  in  D'.  Joining  DD'  and 
producing  gives  on  XX'  the  required  center  5.  Having  found  S, 
it  is  an  easy  matter  to  construct  further  elements  of  K  from  the 
corresponding  elements  of  K\ 


THEORY   OF  CONICS. 


155 


A  similar  problem  was  solved  in  the  first  part  of  this  section. 

5.  Given  three  points  A,  B,  C  of  a  conic  K  and  the  tangents- 
at  A  mid  B.  To  find  the  point  of  intersection  of  this  conic  with 
a  given  line  g. 

Designate  the  intersection  of  the  tangents  by  S,  Fig.  63,  and 
draw  any  circle  K'  tangent  to  SA  and  SB  at  the  points  A'  and 
B'.  K  and  K'  are  now  corresponding  conies  in  a  colHneation 
with  5  as  a  center  and  A,  A';  B,  B'  &s  corresponding  pairs.    SC 


Fig.  63. 


cuts  K^  in  C.  Draw  CA  and  CB  cutting  g  in  X  and  Y.  Draw 
SX  and  SY,  which  by  CM'  and  C'B'  are  cut  in  X'  and  Y\  The 
Hne  joining  X'Y'  is  g'.  Let  g'  cut  K'  in  P'  and  Q',  then  SP' 
and  SQ'  produced  cut  g  in  P  and  Q,  the  points  of  intersection 
of  g  with  K. 

With  exactly  the  same  designations  we  can  immediately 
solve  the  special  case: 

Given  the  asymptotes  of  an  hyperbola  and  another  point.  To 
fi,nd  the  points  of  intersection  of  this  hyperbola  with  a  given  straight 
line. 


^56 


PROJECTIVE  GEOMETRY. 


Nothing  is  changed  in  the  previous  construction  except  that 
A  and  B  are  the  infinite  points  on  the  asymptotes. 

6.  To  construct  a  conic  K  when  three  points  A,  B,  C  and  two 
tangents  a  and  b  are  given. 

Draw  again  a  circle  K'  tangent  to  a  and  h  and  assume  the 
intersection  S  oi  a  and  h  as  the  center  of  coUineation,  Fig.  64. 
Join  S  io  A,  B,  C  and  designate  the  points  of  intersection  of 
SA,  SB,  SC  with  K'  by  A',  B',  C  and  A",  B" ,  C".     If  we  let 


r^acr^ 


Fig.  64. 

A',  B',  C  correspond  to  ^,  B,  C,  then  5  is  the  axis  of  coUineation; 
if  the  corresponding  points  are  A'',  B" ,  C" ,  then  s^  will  be  the 
axis  of  coUineation.  In  both  collineations  the  same  conic  K 
corresponds  to  K' .  But  we  may  also  let  A',  B" ,  C  correspond 
to^,  B,  C,  which  will  lead  to  a  different  conic  K.  The  arrange- 
ment A"B'C",  ABC  leads  to  the  same  conic.  There  are 
eight  different  correspondences  possible  which  in  groups  of 
two  lead  to  the  same  conic.  The  problem  admits,  therejore,  0} 
four  different  solutions. 


THEORY    OF  COMICS. 


57 


7.  Given  three  points  of  a  conic  K, — A,  B,  C, — and  a  focus  S. 
To  construct  the  conic. 

Draw  in  Fig.  65  any  circle  tangent  to  the  conjugate  imaginary- 
tangents  from  S  to  K;  i.e.,  draw  anv  circle  K'  with  5  as  a  center. 


Fig.  65. 

The  problem  and  the  construction  are  in  this  case  exactly  the 
same  as  in  problem  6.     Here  also  there  are  four  different  solutions. 

8.  Given  four  points  and  a  tangent  of  a  conic,  to  construct  it. 

In  Fig.  66  let  A,  B,  C,  D  he  the  given  points  and  t  the  given: 
tangent  of  the  conic  K.  Consider  AB  or  s  siS  the  axis  of  a  col- 
lineation,  and  any  circle  K'  through  A  and  B  as  the  perspective 
of  K.  By  means  of  the  quadrilateral  A  BCD  construct  the  polar 
p  oi  P  with  respect  to  K,  and  also  the  polar  p'  of  P  with  respect 
to  K\  p  and  p'  meet  in  a  point  of  5.  From  the  pomt  where  t 
cuts  5  draw  the  tangent  t'  \o  K'  and  let  this  tangent  correspond 
to  /  in  the  collineation.  t  and  t '  cut  p  and  p'  in  two  correspond- 
ing points  X  and  A^',  and  the  center  or,  if  there  are  several,  the 
centers  of  possible  collineations  must  be  on  the  line  joining  X 
with  X' .  To  determine  these  centers,  join  C  with  X,  and  the 
point  of  intersection  of  CX  with  5  to  X' .     Where  the  last  Une  cuts 


158 


PROJECTIVE   GEOMETRY. 


K'  are  two  points  C  and  C/  which  correspond  to  C.     Hence  there 
are  two  different  collineations.     Their  centers  5  and  S^  are  ob- 


FiG.  66. 


tained  as  intersections  of  CC  and  C^C  with  XX'.     To  the  point  of 
tangency  T  ol  t'  with  K'  correspond  in  the  two  colhneations  T 


THEORY   OF  CONICS.  159 

arid  Ty^  on  /,  which  are  the  points  of  tangency  of  the  two  conies  K 
passing  through  A  BCD  and  tangent  to  /. 

That  these  are  the  only  solutions  is  not  apparent  from  this 
construction;  it  simply  shows  how  conies  with  the  required  con- 
ditions may  be  found. 

9.  Osculating  Circle  of  a  Conic. 

If  a  conic  K  passes  through  the  center  5  of  the  collineation, 
then  K'  will  be  tangent  to  K  at  5.  If,  furthermore,  also  the  axis 
^  passes  through  S,  one  of  the  remaining  two  points  of  intersection 
of  K  and  K'  will  coincide  with  S,  and  K  and  iv'  have  at  5  a  con- 
tact of  the  second  order.  If  K  is  a  circle,  K  will  be  the  osculat- 
ing circle  to  K'  at  S.  The  remaining  fourth  point  of  intersection 
will  be  on  s.  In  case  that  5  is  on  s,  the  counter-axes  of  collinea- 
tion g'  and  r  will  be  on  opposite  sides  of  5.  The  center  AI  of  the 
circle  K  is  the  pole  of  the  infinitely  distant  line  q  with  respect  to 
K.  The  corresponding  point  M'  of  the  collineation  is  the  pole  of 
g'  with  respect  to  K'.  If  now  a  diameter  of  K  turns  about  M, 
the  rays  joining  5  with  its  extremities  form  a  rectangular  involu- 
tion of  rays  around  5  which  is  identical  with  the  involution  of  rays 
joining  5  to  the  extremities  of  the  chords  through  M^  correspond- 
ing to  the  diameters  through  M.  Hence  if  is  obtained  as  the 
pole  of  the  involution  of  points  on  K',  which  when  joined  with  5 
give  a  rectangular  involution  of  rays.  5'  is  the  polar  of  M'  with 
respect  to  K',  and  5  is  a  line  through  S  parallel  to  q'. 

It  is  now  possible  to  solve  the  problem :  Given  five  points  of  a 
conic  K' ,  to  construct  the  osculating  circle  at  any  of  the  given 
points. 

In  Fig.  67  let  A,  B,  C,  D,  E  be  the  given  points,  and  A 
the  point  at  which  the  osculating  circle  shall  be  constructed. 
Join  A  with  B  and  C;  Sit  A  erect  perpendiculars  to  AB  and  A  C, 
and  by  Pascal's  theorem  construct  the  intersections  B^  and  C^ 
of  these  perpendiculars  with  K'.  BB^  and  CC^  cut  each  other 
at  M'.  In  the  quadrilateral  BB^CC^  the  polar  q'  of  M'  is  easily 
found.  Through  A  draw  5  parallel  to  q',  and  find  by  Pascal's 
theorem  the  intersection  F  of  s  with  K'.  K  is  the  circle  passing 
through  F  and  tangent  at  A  to  the  perpendicular  to  AM\ 


i6o 


PROJECTIVE   GEOMETRY. 


If  K  is  tangent  to  s  at  the  point  S,  the  osculation  will  be  of 
the  third  order;  i.e.,  K  and  K'  will  have  four  points  in  common 
at  5. 

Ex.  I.  To  construct  a  conic  when  the  osculating  circle  at 
one  of  its  points,  A,  and  two  other  points,  B  and  C,  are  given. 


Fig.  67. 


Ex.  2.  To  construct  the  osculating  circle  at  the  vertex  of  a 
conic  which  is  determined  by  major  and  minor  axes. 

Ex.  3.  To  construct  a  parabola  when  the  osculating  circle  at 
its  vertex  is  known. 

Ex.  4.  Given  three  tangents  and  two  points  of  a  conic;  con- 
struct the  conic.     Dualistic  problem  of  §  42,  6. 

Ex,  5,  Given  four  tangents  and  a  point  of  a  conic;  construct 
the  conic.     Duahstic  problem  of  §  42,  8. 


THEORY   OF   CONICS.  l6l 


§  43.   Problems  of  the  Second  Order. 

I.  In  the  previous  section  and  even  farther  back  we  have 
occasionally  touched  upon  problems  of  the  second  degree.  We 
shall  now  pay  particular  attention  to  a  few  geometrical  problems 
which  analytically  are  equivalent  with  the  solution  of  an  equation 
of  the  second  degree.  Most  of  these  problems  may  be  reduced 
to  the  problem,  to  find  the  double-points  of  two  coincident  projective 
point-ranges.  This  was  done  analytically  in  the  first  chapter. 
Geometrically  we  may  solve  it  by  the  following  proposition,  which 
in  a  little  different  form  appears  as  Pascal's  theorem :  Six  points, 
A,  B,  C,  A',  B' ,  C ,  on  a  conic  K  determine  two  projective  ranges 
of  points  on  K,  so  that  for  any  point  P  on  i^  we  have  the  pro- 
jective pencils 

P{ABC  .  .  .)  =  P{A'B'C' .  .  .). 

The  pairs  of  sides  AB',  A'B;  BC,  B'C;  CA',  CA  intersect  in 
three  collinear  points,  on  the  Pascal  line  p.  Considering  two 
points,  for  instance  B  and  B' ,  as  carriers  of  pencils,  then 

{BA'B'C  ..  .)  =  {B'-ABC  ...), 

and  as  BB'  is  a  ray  common  to  both  pencils,  they  are  perspective 
and  have  p  as  the  axis  of  perspective.  Two  rays  joining  B  and 
B'  with  any  point  on  p  cut  K  in  two  corresponding  points  of  the 
projective  ranges  on  K.  From  this  it  is  clear  that  the  points  of 
intersection  of  p  with  K  are  the  double-points  of  the  projectivity. 
These  are  real,  coincident,  or  imaginary  according  as  p  cuts, 
touches,  or  does  not  cut  K. 

If,  instead  of  six  points,  six  tangents  a,  b,  c,  a',  b' ,  c'  of  the 
conic  are  given,  the  lines  joining  the  pairs  of  intersection  ab' , 
a'b;  be',  b'c;  ca' ,  c'a  all  pass  through  the  Brianchon  point  P. 
Considering  tv/o  of  the  tangents,  b  and  b' ,  and  cutting  these  with 
the  remaining  tangents,  two  perspective  ranges 

{b-a'b'c'  ...)  =  {b'-abc...) 


l62 


PROJECTIVE   GEOMETRY. 


with  the  common  point  hh'  (self- corresponding)  are  obtained. 
Taking  any  ray  through  P  and  cutting  it  by  h  and  V ,  then  the 
tangents  to  K  from  these  two  points  are  two  corresponding  tan- 
gents. The  tangents  from  P  to  K  are  the  double  tangents  of  the 
projectivity.  We  postulate  now-  that  a  problem  is  geometrically 
solvable  if  it  can  be  solved  by  compass  and  ruler.  Hence,  replac- 
ing K  by  a,  circle,  we  can  now  solve  the  following  problems: 


Fig.  68. 


2.  Given  five  points  of  a  conic;  to  construct  the  intersections 
of  this  conic  with  a  straight  line. 

In  Fig.  68  let  A^B^C^D^E^  be  the  given  points  and  /  the 
given    line.      Join  D^  and  E^  with  A^B^C^  and  designate    the 


t 


THEORY    OF  CONICS. 


163 


corresponding  intersections  on  /  by  ABC  and  A'B'C.  Draw 
an  arbitrar}^  circle  A'  and  join  any  of  its  points  5  with  ABC  and 
A'B'C ,  and  let  these  lines  cut  A'  in  points  which  we  shall  also 
designate  by  ABC  and  A'B'C,  to  simplify  the  designation. 
According  to  the  foregoing  results  ABC  and  A'B'C  determine 
two  projective  ranges  on  A.  Construct  the  line  p  and  let  M 
and  N  be  the  intersections  of  p  with  A.  Join  5  with  M  and  N 
on  A,  and  produce  to  their  like-named  intersections  M  and 
N   on   l.      These    will  be   the   double-points   of    the   projective 


Fig.  69. 

ranges  ABC  .  .  .  and  A'B'C,  or  the  points  of  intersection  of  / 
with  the  conic  through  ABCDE. 

3.  Given  five  tangents  ■  of  a  conic;    to  construct  the  tangents 
vj  this  conic  through  a  given  point. 


1 64  PROJECTIVE   GEOMETRY. 

In  Fig.  69  let  a^,  b^,  c^  di,  e^  be  the  given  tangents  and  5  the 
given  point.  Cut  d^  and  e^  with  aJ)^Cj  and  designate  the  corre- 
sponding Hnes,  joining  these  points  with  5",  by  a  bc^  a'  b'  c'.  Draw 
an  arbitrary  circle  K  and  let  any  tangent  s  oi  K  cut  abca'b'd  in 
six  points.  From  these  draw  tangents  to  K  and  designate  them 
similarly  by  abc,  a'b'c' .  Construct  the  Brianchon  point  F  of  the 
circumscribed  hexagon  ahca'b'c'  of  K.  From  F  draw  the  tan- 
gents m  and  n  to  -K",  cutting  5  in  M  and  N .  The  lines  joining  5 
with  M  and  N  are  the  required  tangents  from  5  to  the  given 
conic. 

This  construction  may  be  replaced  by  a  simpler  one.  Join 
the  points  w^here  a^,  &i,  c^  cut  d^  and  e^  directly  to  the  point  5  of 
an  auxiliary  circle  passing  through  5,  and  designate  the  points 
of  intersection  with  this  circle  by  ^,  ^,  C,  ^',  5',  C .  Con- 
struct the  Pascal  line  p  of  this  hexagon.  The  lines  joining  5  with 
the  points  of  intersection  of  -p  with  the  auxiliary  circle  are  the 
required  tangents  from  5, 

4.  Poncelet's  Problem. — To  construct  a  polygon  whose 
vertices  shall  lie  on  given  straight  lines  {each  on  each),  and  whose 
sides  shall  pass  through  given  points  {each  through  each). 

For  the  sake  of  simplicity  we  shall  limit  the  problem  to  four 
straight  lines  a,  b,  c,  d  and  four  points  ^,  B,  C,  D.  The  method 
of  reasoning  is  not  different  in  the  general  case.  First  make  a 
trial  construction  by  drawing  through  A  a  line  0^  cutting  a  m  A^y 
Fig.  70.  From  A^  draw  a  line  b^  through  B,  cutting  b  in  B^; 
from  B^  a  line  c^  through  C,  cutting  c  in  C^;  from  Cj  a  line  d^ 
through  D,  cutting  dinD-^^',  from  D^  a  line  a/  through  A  cutting 
a  m  A^.  If  a^  turns  about  A,  then  b^,  c^,  d^,  a/  will  turn  about 
B,  C,  D,  A  in  such  a  manner  that  we  have  for  various  positions 
the  projective  ranges 

{A,A,A, .  .  .)  =  {B,B,B, .  .  .)  =  (CAC3  •  •  0  = 

(AAA...)  =  (^/^/-4/...). 

Considering  the  projective  ranges 

(A,A,A,...)=^{A/A,'A,\..) 


THEORY   OF   CONICS. 


If>5 


on  a,  it  is  clear  that  the  hnes  a,^,  a^  through  A  and  the  double- 
points  of  these  ranges,  M  and  N,  couicide  with  the  lines  aj, 
aj  through  D.  These  double-points  determine,  therefore,  two 
solutions  of  the  problem  which  may  be  real  or  imaginary.     In 


the  figure  the  two  real  solutions  of  the  problem  are  indicated  by 
heavy  lines. 

Ex.  I.  To  inscribe  in  a  given  conic  a  polygon  whose  sides 
pass  respectively  through  given,  non-collinear,  points. 


1 66  PROJECTIVE  GEOMETRY. 

Ex.  2.  To  circumscribe  about  a  given  circle  a  triangle  whose 
vertices  are  on  three  given  lines. 

Ex.  3.  Betv^een  two  given  straight  lines  to  place  a  segment 
such  that  it  shall  subtend  given  angles  at  two  given  points. 

Ex.  4.  To  construct  a  polygon  whose  sides  shall  pass  respect- 
ively through  given  points,  and  all  whose  vertices  except  one 
shall  lie  respectively  on  given  straight  lines;  and  which  shall  be 
such  that  the  angle  included  by  the  sides  which  meet  in  the  last 
vertex  is  equal  to  a  given  angle.     (Cremona.) 

Let  A,  B,  C,  .  .  .  ,  N  be  the  given  points  and  a,  b,  c,  .  .  .  ,  m 
the  given  lines.  Through  A  and  N  draw  a  circle  K  which  sub- 
tends the  given  angle  over  the  chord  AN.  From  any  point  of  K 
draw  a  line  through  A,  cutting  a  in  A^;  from  A^  draw  a  line 
through  B,  cutting  b  in  B^,  and  so  forth,  until  the  line  m  is  reached 
in  a  point  M^.  Then  through  N  and  the  same  point  on  K  draw 
a  line  cutting  m  in  M/.  Repeat  this  construction  for  two  other 
points  of  K,  thus  giving  on  m  the  projective  ranges  (yl  1^43^3  .  .  .)  = 
{A  i' A  2' A  3  .  .  .).  The  double-points  of  these  ranges  make  it 
possible  to  draw  two  polygons  with  the  required  properties. 
This  problem  may  be  solved  in  a  different  manner. 

Through  A  draw  any  line  a^  cutting  a  in  A^,  through  A  ^  and 
B  a  line  ^j,  and  so  forth,  until  the  line  m  is  reached  in  M^. 
Through  M^  and  N  draw  a  line  n^  cutting  a^  in  a  point  V.  If 
now  a^  turns  about  A,  then  n^  will  turn  projectively  about  N. 
Hence  their  point  of  intersection  V  will  describe  a  conic  K* 
passing  through  A  and  N.  The  two  other  points  of  intersection 
of  this  conic  with  the  circle  K  determine  the  two  solutions  of 
the  problem.  It-  is  now  possible  that  the  conic  K*  is  itself  a 
circle,  but  different  from  K.  In  this  case  there  is  no  real  solu- 
tion. X*  may  be  identical  with  K,  so  that  there  are  an  infinite 
number  of  solutions. 

Make  the  constructions  as  indicated. 


THEORY  -OF   CONICS. 


167 


§  44.   An  Optical  Problem. 

I.  The  following  problem  stated  by  Cremona  in  his  Elements 
of  Projective  Geometry,  p.  199,  is  an  application  of  Ex.  4  of  the 
previous  section: 

A  ray  of  light  emanating  jrom  a  given  point  A  is  reflected 
jrom  n  given  straight  lines  in  succession;  to  determine  the  original 
direction  which  the  ray  must  have,  in  order  that  this  may  make 
with  its  direction  after  the  last  reflection  a  given  angle. 


Designate  in  Fig.   74  the  reflecting  lines  by  a,  b,  c, .  . . ,  n. 
Through  A   draw  any  ray  a^  striking  a  at  A^.     The  reflected 


1 68  PROJECTIVE  GEOMETRY. 

ray  h^  passes  through  the  point  B  which  is  symmetrical  to  A 
with  respect  to  a.  The  ray  h^  strikes  h  at  -Sj,  and  its  reflected 
ray  c^  passes  through  C,  which  is  symmetrical  to  B  with  respect 
to  h,  and  so  forth.  The  ray  o^  reflected  from  the  last  line  n 
at  Ni  passes  through  O,  which  is  symmetrical  to  N  with  respect 
to  n.  Let  the  rays  a^  and  o^  intersect  at  \\.  We  have  now  a 
closed  polygon  a^h^c^ .  .  .  o^,  whose  sides  pass  through  the  fixed 
points  A,  B,  C,  .  .  .  N,  O  and  whose  vertices  except  V  lie  on  the 
fixed  sides  a,b,c,...,n.  Hence,  when  a^  turns  about  A ,  V  will 
describe  a  conic  and  the  problem  is  reduced  to  the  one  explained 
in  Ex.  4,  §  43. 

2.  Cremona  stops  the  discussion  of  the  problem  at  this  point. 
We  shall  now  show  that  a  further  investigation  is  necessary. 
Let  a  be  the  angle  of  incidence  of  a^  on  a,  a^  the  angle  of  inci- 
dence of  ^1  on  b,  a^  of  c^  on  c,  and  so  forth;  ^j,  cf)^,  (f>3  ■  •  ■  the  angles 
which  a  and  b,  b  and  c,  c  and  d,  .  .  .  include.  The  angles  (f>i,  (p2, 
^3,  .  .  .  between  the  different  reflecting  lines  must  be  selected  in 
such  a  manner  that  always 

From  the  figure  we  now  derive  the  following  series: 

«!=       a, 

cx.2  =  7z—a—  cf)y, 

a:g  =  7r— a—  <55)i+  (l>2—  ^3+  ^4-  ^5» 


a^^+i  =       a+  </>i—  (f)2+  (j)s—  ...  —  02/i- 

If  there  are  n  reflecting  lines,  then  the  number  of  angles  a 
is  also  n,  and  the  number  of  angles  <f)isn—i.     Erect  perpendicu- 


THEORY   OF  CONICS.  i6g 

lars  to  a^  at  any  of  its  points,  to  a  at  ^  i,  to  h  at  B^,  and  so  forth, 
to  n  at  any  of  its  points.  Then  the  first  and  the  last  perpen- 
dicular deviate  from  each  other  by  the  angle 

a+  {n-  2)7Z+an-  (<^i+  02+  ^3+   .  .  .  +  (f^n-l), 

and  consequently  also  the  rays  a^  and  o^  by  the  angle  0: 
0=a  +  Q:„— (01+02+03+  .  .  .  +0n-i)+(«-2)7r. 
If  n  is  odd,  then  n—  i  is  even;  i.e.,  w  =  2^+  i,  and 

0  =a:  +  Q;+  01—  02+  03—  ...—  03^;—  0^—  03—  03—  ...—  02/X+  (W—  2)71, 

or 

0  =  (w-  2)7r+  2a-  2(02+  02+  04+  ...  +  02/.) ; 

i.e.,  the  angle  between  the  original  incident  ray  of  Hght  and  the 
final  emanant  ray  depends  upon  the  angles  0  and  the  original 
angle  of  incidence  a.  Their  point  of  intersection  V  describes  a 
conic  which  is  not  a  circle,  and  there  are  two  positions  of  F,  real 
or  imaginary,  for  which  the  incident  and  reflected  ray  make  a 
given  angle. 

If  n  is  even,  then  n—  i  is  odd;  i.e.,  n  =  2fx  and 

0=a+;r -a -01+02-03+. ..-0,/,_i-0i-02-... -02^,-1+ (w-2);t, 


or 


0  =  {n-  l)7r-  2(01+  03+  ...  +  02;,_i). 


Hence,  in  case  of  an  even  number  of  reflecting  sides,  V  describes 
a  circle  and  the  angle  0  is  constant.  Cremona's  problem  admits 
either  of  no  solution,  or  of  an  infinite  number  of  solutions.  The 
angle  0  does  not  depend  upon  the  angles  0  of  even  indices. 

To  sum  up  we  may  state  Cremona's  problem  and  its  solution 
by  the  following  proposition: 

//  rays  of  light  emanate  from  a  fixed  source  which,  in  succession, 
are  reflected  on  n  straight  lines,  then  the  last  reflected  rays  cut  the 


170 


PROJECTIVE  GEOMETRY. 


corresponding  original  rays  in  points  of  a  conic,  which  is  not  a 
circle,  when  n  is  odd,  and  in  points  of  a  circle  when  n  is  even.    In 


Fig.  72. 

the  first  case  there  are  two  places  on  the  conic  at  which  the  original 
and  the  final  ray  make  a  given  angle.  In  the  second  case  there  are 
no  such  places  on  the  circle,  or  else  an  infinite  number.    In  this 


\9=W 


Fig.  73. 

case  the  angle  cj)  depends  only  upon  the  angles  between  succeeding 
reflecting  lines  whose  orders  in  this  succession  are  odd. 
3.  Applications.^ — Let  n  =  2,  then  (p=7i—2<f>^. 


To  make  ^=— ,  we  must  choose  ^^^ 


=  45°.     This  case  is  illus- 


THEORY  OF   CONICS.  171 

trated  in  Fig.  72  and  is  practically  applied  in  Bauemfeind's  Angle 
Mirror  or  Optical  Square. 
For  w  =  4 

To  make  (f>  =—,  9^1+9^3  must  be  made  equal  to —tt.  Under 
this  condition  (p^  and  (563  may  vary  separately.  The  condition 
is  also  satisfied  by  taking  96i  =  <^2  =  F^=ii2°  30',  and  this  is 
illustrated  in  Fig.  73. 


CHAPTER  IV. 

PENCILS  AND  RANGES   OF  CONICS.      THE  STEINERIAN  TRANS- 
FORMATION.     CUBICS. 

§  45.   Pencils  and  Ranges  of  Conies. 

I.  Involution  of  the  Pencil  #+A2f/^)  =  o. 

Let  u=ax^-\-2bxy-\-cy^-\-2dx-\-2ey+f  =  o, 

Ui  =  aj^x^+  2b^xy-\-c{y'^-\-  2d^x^  2gi^'+/i  =  o 

be  the  equations  of  two  conies;   then 

(i)  W+AMi  =  o 

is  the  equation  of  a  conic  which  passes  through  the  four  points 
of  intersection  of  V  and  V^.  As  a  conic  is  determined  by  five 
points,  any  fifth  point,  different  from  one  of  the  four  points  of 
intersection  of  Z7  and  f/^,  determines  the  equation  of  the  conic 
through  the  five  points;  i.e.,  X.  Conversely,  every  value  of  X 
determines  the  equation  of  one  of  the  conies  of  the  system. 
Designating  the  points  of  intersection  of  V  and  [/"i  by  ^,  B,  C,  D, 
then  for  a  variable  X, 

u+Xu^  =  o 

represents  all  conies  through  the  four  fixed  points  A,  B,  C,  D, 
and  is  called  the  equation  of  the  pencil  of  conies  through  these 
points.     Among  these  conies  are  three  degenerate  conies,   con- 

^  See  Joachimsthal,  loc.  cit.,  p.   183.     By  U,  V,  Ui,  etc.,  we  shall  designate 
conies  whose  equations  are  m=o,  v=o,  u^^o,  etc. 

The  student  is  asked  to  draw  a  figure  for  this  section. 

172 


PENCILS  AND  RANGES  OF  CONICS.  IJS 

isting  of  the  three  pairs  of  hnes  through  A,  B,  C,  D.     To  prove 
this  we  form  the  discriminant  of  (i),  which  is 


(2) 


a+Xa^  b-\-Xbi  d-{-)d^ 
h+Xh^  c+Aq  e+Ae^ 
d+Xd^       e+Xe^        /+A/i 


The  vanishing  of  this  expression  is  the  condition  for  degener- 
ate conies  among  the  pencils.  This  gives  a  cubic  equation  in  X 
and  consequently  three  values  for  A;  i.e.,  three  degenerate  conies 
through  A,  B,  C,  D,  as  was  to  be  proved.  One  value  of  X  is 
always  real,  so  that  also  in  case  of  one  or  two  imaginary  pairs 
among  A,  B,  C,  D  there  is  always  a  conic  consisting  of  a  real 
line-pair.  In  case  of  a  double- root  which  is  evidently  real,  the- 
third  root  is  also  real;  the  conies  U  and  U^  have  a  contact  of 
the  first  order.  If  the  root  is  triple,  U  and  U^  have  a  contact  of 
the  second  or  third  order. 

In  §  36  it  was  shown  that  the  coordinates  of  a  point  C  (x,  y) 
on  the  line  joining  the  two  points  A(x^,  yj},  B^x^,  y-^)  are 

x^—Xx^  y^—Xy^  AC 

x  = r^,     v= p,     where      A  =  ^7;?. 

I  — /  I  —  /  BL 

Assume  now  that  A   and  B  are  on  the  conic  given  by  (i), 

AC 
then  C  is  on  the  conic  Z7,  if  A  =  -^  is  a  root  of 

(3)  Mj— 2Az;+A^W2  =  o, 

where  w^,  v,  u^  have  the  same  meaning  as  in  formula  (2),  §  36. 
In  a  similar  manner,  C  is  on  U^,  if  X  satisfies 

(4)  u/—2Xv'+X^U2=o, 

where  u/,  v' ,  u^    have  the  same  meaning  as  in  (3),  except  that 
a,  &,  c,  .  .  .  are  replaced  by  a^,  b^,  q,  .  .  .  .     Designating  the  points 


174  PROJECTIVE  GEOMETRY. 

of  intersections  oi  AB  produced  with  U  by  C,  C'\   and  with  U^^ 
by  Ci,  C/,  we  have 

AC  AC__u^     AC,  AfJ^jal 
BC'BC'~u^'    BC^'BC^^u'f 

The  points  A  and  B  are  on  m+Ami  =  o;  hence 

and,  eliminating  A, 

—  =  —7,     or 


(5) 


A£Aa_AC,  AC  I 
BC'BC'~  BC^BCr 


Giving  \  all  possible  values  and  keeping  the  transversal, 
or  C,  C  and  Ci,  C/,  fixed,  ^,  J5  is  the  pair  of  points  in  which 
the  variable  conic  u-\-hi^  =  Q  cuts  this  transversal.  (5)  may 
also  be  written 

{ABCC;)  =  {BAC'Cl)\ 

i.e.,  the  anharmonic  ratio  of  any  four  points  cut  out  by  the  fixed 
and  variable  conies  on  the  transversal  is  equal  to  the  anharmonic 
ratio  of  the  four  corresponding  points.  Furthermore,  from  (5) 
it  is  seen  that  interchanging  A  and  B,  two  corresponding  points, 
does  not  affect  the  relation.  The  system  of  points  defined  by  (5) 
is  therefore  involutoric.     Hence  the  theorem: 

The  conies  of  a  pencil  of  conies  cut  any  transversal  in  an  involu- 
tion of  points.  Every  conic,  including  the  degenerate  conies,  cuts 
out  a  pair  of  the  involution.     (Desargues.) 

The  double-points  of  the  involution  are  evidently  the  points 
where  two  conies  of  the  pencil  touch  the  transversal.  They  may 
be  real   (including  coincidence)  or  imaginary.      The  remark  in 


PENCILS  AND   RANGES  OF  CONICS.  175 

connection  with  problem  8,  §  42,  to  construct  a  conic  through 
four  given  points,  tangent  to  a  given  hne,  is  now  clear. 

Corollary. — Any  transversal  cuts  the  three  pairs  of  sides 
of  a  complete  quadrilateral  in  three  pairs  of  an  involution. 

Ex.  Prove  directly  that  every  transversal  cuts  a  coaxial  system 
of  circles  in  an  involution.  By  reciprocation  we  derive  the 
theorem : 

The  pairs  of  tangents  from  any  point  to  the  conies  of  a  range 
{conies  inscribed  in  a  quadrilateral)  form  an  involutoric  pencil. 

Corollary. — The  lines  joining  any  point  with  the  three 
pairs  of  vertices  of  a  complete  quadrilateral  form  three  pairs  of 
an  involutoric  pencil. 

Ex.  Prove  directly  that  the  tangents  from  any  point  to  the 
range  of  circles  inscribed  to  two  straight  lines  form  an  involutoric 
pencil. 

2.  A  Special  Case. — Assume  the  four  points  A,  B,  C,  D, 
through  which  the  pencil  of  conies  passes,  as  an  orthogonal  quadri- 
lateral; i.e.,  ABA.CD,  BC  LAD,  CA  ±BD.  In  this  case  the 
conies  are  all  equilateral  hyperbolas.  To  prove  this,  note  that  the 
degenerate  conies  consist  of  three  pairs  of  perpendicular  lines, 
Fig.  74.  The  involution  on  the  infinitely  distant  line  is  there- 
fore rectangular  and  its  pairs  can  only  be  cut  out  by  conies 
whose  infinite  branches  are  rectangular;  i.e.,  branches  of  equi- 
lateral hyperbolas. 

Take  now  any  equilateral  hyperbola  and  on  it  any  triangle 
ABC.  Let  D  be  the  point  of  concurrence  of  the  altitudes  of 
ABC.  Through  A  BCD  we  can  now  pass  an  infinite  number 
of  equilateral  hyperbolas,  among  which  is  necessarily  the  given 
hyperbola.  Hence  D  is  on  this  hyperbola,  and  we  have  the 
theorem : 

The  point  of  concurrence  of  the  altitudes  of  any  triangle  in- 
scribed in  an  equilateral  hyperbola  lies  on  this  hyperbola.^ 

3.  Polar s  of  a  Pencil  of  Conies. 


*  Brianchon  et  Poncelet  in  Gergonne's  Annales,  Vol.  II,  pp.  205-220.     Also 
Fiedler  in  Vierteljahrsschrift  d.  Naturf.-Ges.,  Ziirich,  Vol.  XXX,  pp.  390-402. 


176  PROJECTIVE   GEOMETRY. 

From  the  explicit  expression  of  the  equation  of  a  pencil  of 


comes 


(6) 


u-\-Xu^  =  o 


it  is  easily  found  that  the  equation  of  the  polar  of  a  point  P  may 
always  be  put  in  the  form 


(7) 


p+Xp,  =  o, 


where  p  =  o,  pi  =  o  are  the  equations  of  the  polars  of    P  with 
respect  to   the  conies  U  and  U^.     From  this  it  follows  that  the 


Fig.  74. 

polar  of  any  point  P  with  respect  to  a  conic  of  the  pencil  always 
passes  through  the  point  of  intersection  P'  of  the  polars  of  P 
with  respect  to  U  and  U^.     Hence  the  theorem : 

All  polars  of  a  point  with  respect  to  the  conies  of  a  pencil  are 
concurrent. 

If  the  point  P  with  the  coordinates  x^,  y^  describes  the  straight 


PENCILS  AND   RANGES  OF  CONICS. 


177 


line  aXi+/?}'i+^=o,   then  for  the  point  P'={x^y^)   we   have 
the  three  conditions 

(ax/  J-  &)'/  +  d)x^  +  (6x/  +  c^'/  +  e)};i  +  t^x/  +  ey^  +  /  =0, 

(a^x/  +  &!>'/  +  (/Jxi  +  (&ix/  +  c^y^  +  O^'i  +  d^x^  +  ^i)//  +  /i  =  o,, 

ax^+  ^y^+  X  =0, 

which  are  consistent  only  when 


(8) 


ax/  +by/  -\-d      bx^  +cy/  +e     dx/  +ey/  +  f 
a^x/  +  &i3'/  +  di     h^x/  +  c^y/  +  e^     d^x/  +  eji'  +  /^ 


This  gives  a  quadratic  equation  between  x/,  }»/,  the  coor- 
dinates of  P' ;    hence  the  theorem: 

//  a  point  P  describes  a  straight  line,  then  the  point  of  con- 
currence P'  of  all  polars  with  respect  to  the  conies  of  a  pencil 
describes  a  conic. 

Designating  ax-^'  +  by/  -\-d,  J^Z  +  cvZ+f,  dv/+g)'/+/  by  r,  s,  t, 
and  in  a  similar  manner  by  r^,  s^,  t^  the  same  expressions,  v/ith 
a,  b,  e,  .  .  .  replaced  by  a^,  b^,  q,  the  polar  of  a  point  (x/,  y/)  with 
respect  to  the  pencil  of  conies  u-{-Xu^  =  o  has  the  form 

(r^  Xr^)Xi+  {s+  Xs^)y^  +  t+  Xti  =  o. 

This  equation  will  be  identical  with  that  of  the  given  line  g, 

«^i+/5>'i+r=o,     if 

r+Xr^     s-\-Xs^    t+Xt^ 


(9) 


/? 


Hence  the  pole  (x/,  y/)  of  g  for  any  conic  of  the  pencil  must 
satisfy  (9).  For  every  value  of  A  a  definite  value  of  (x/,  y/) 
is  obtained,  which  therefore  describes  a  certain  locus.     To  find 


I  78  PROJECTIVE   GEOMETRY. 

its  equation  we  must  eliminate  X  from  (9),  which  gives  the  equa- 
tion 

(10)  a{sti-  s^t)+^{r4-rQ  +  ri^h-  r^s)  =  o. 

But  this  is  identical  with  (8).     Hence  the  theorem: 
The  locus  of  the  poles  0}  a  straight  line  with  respect  to  all 
conies  of  a  pencil  is  a  conic  which  is  identical  with  the  conic  of 
concurrent  polars  of  all  points  of  the  given  line  with  respect  to 
the  same  pencil. 

Of  particular  interest  is  the  case  when  P  describes  the  infi- 


y 

I.e.,  wnen 
X,  =  00.     In  this  case 


nitely  distant   line;   i.e.,  when  —  =  /i  is   a   variable    (finite)    and 


ip=  ax/  +  by/  +d  +  n(bx/  +  cy/  +  e  )  =0, 
^^  \  p^  =  a^x/  +  hj/  +  d^-\-  fx(b^x/  +  c^y/  +  e^)  =0. 

Eliminating  fi,  the  equation  of  the  conic  which  P'  describes 
becomes 

(i  2)     (ax/  +  by/  +  d)  (b,x/  +  cj/  +  e/) 

-  (a^x/  +  bj/  +  d/)  (bx/  +  cy/  +  e)  =  o . 

If  (Xj,  y/)  describes  the  infinitely  distant  line,  then  its  pole 
with  respect  to-  a  certain  conic  of  w+/iWj  =  o  must  satisfy 
p+Xp^  =  o  for  all  values  of  //,  which  can  only  be  true  when  equa- 
tion (12)  is  satisfied.  But  the  poles  of  the  infinite  line  are  the 
middle  points  of  the  conies  of  the  pencil.  The  centers  of  a  pencil 
of  conies  lie  on  a  conic  whose  equation  is  given  by  (12). 

The  three  diagonal  points  of  the  fundamental  quadrilateral^ 
being  the  centers  of  the  three  degenerate  conies,  belong  to  this 
locus.  If  P  is  taken  as  the  infinitely  distant  point  of  the  fine 
joining  two  points,  say  A  and  B,  of  the  fundamental  quadri- 
lateral, then  P'  is  the  middle  point  of  A  B,  since  all  polars  of  P 
with  respect  to  the  conies  of  the  pencil  pass  through  this  point. 
The  locus  (12)  passes,  therefore,  also  through  the  middle  points 


PENCILS  AND   RANGES  OF   CONICS.  179 

of  AB,  BC,  CD,  DA,  BD,  CA.  In  case  of  an  orthogonal  quad- 
ruple, as  it  was  described  above,  under  (2),  the  locus  (12)  becomes 
a  circle  circumscribed  to  the  foot-points  of  the  altitudes  of  the 
triangle  ^^C,  which  bisects  the  sides  and  the  segments  DA,  DB, 
DC  of  the  altitudes.  Fig.  74.  This  circle  is  otherwise  called 
the  Feuerbach  circle  ^  of  the  triangle.  We  have  therefore  the 
theorem : 

The  locus  of  the  centers  of  all  equilateral  hyperbolas  circum- 
scribed to  an  orthogonal  quadrilateral  A  BCD  {D= point  of  concur- 
rence of  altitudes)  is  the  Feuerbach  circle  of  the  triangle  ABC. 

4.  Poles  of  a  Range  of  Conies. 

A  range  of  conies  consists  of  all  conies  inscribed  to  a  quadri- 
lateral (imaginary  elements  included),  or  is  the  reciprocal  of 
a  pencil  of  conies  u-\-Xu^  =  o  with  respect  to  a  given  conic  K. 
From  this  property  we  derive  immediately  the  theorems : 

A II  poles  of  a  straight  line  with  respect  to  the  comes  of  a  range 
are  colUnear. 

If  a  straight  line  p  turns  about  a  fixed  point,  then  the  line  of 
collinearity  p'  of  all  its  poles  with  respect  to  the  comes  of  a  range 
envelops  a  conic. 

Let  M  be  the  center  of  K  and  designate  by  F  a  conic  of 
the  pencil  m+Aw^=o,  and  by  v  the  polar  of  M  with  respect 
to  V.  On  reciprocation  with  respect  to  K,  V  is  transformed 
into  a  conic  V ;  M,  the  pole  of  v,  is  transformed  to  infinity; 
consequently  the  polar  v  is  transformed  into  the  center  of  the 
transformed  conic  V.  As  the  polars  of  M  with  respect  to 
all  conies  V  are  concurrent,  it  follows  that  their  reciprocal  poles  are 
colUnear. 

Hence  the  theorem: 

The  centers  of  the  conies  of  a  range  are  colUnear. 

Among  the  conies  of  the  range  there  are  three  degenerate 
ones,  consisting  of  the  three  pairs  of  points  in  which  the  sides 
of    the    fundamental    quadrilateral    intersect    each    other.     The 

^  Concerning  this  circle  see  Cajori's  History  of  Elementary  Mathematics,  pp. 
259,  260;  also  Kotter:  Die  Entwickelung  der  synthetischen  Geometric,  Vol.  I, 
PP-  35-38- 


l8o  PROJECTIVE  GEOMETRY. 

middle  points  of  these  three  pairs  evidently  belong  to  the  above 
lociis.     This  may  be  stated  in  the  corollary: 

The  middle  points  oj  the  three  diagonals  oj  a  complete  quadri- 
lateral are  collinear. 

Designating  the  three  pairs  of  points  by  A-^A^,  B^B^,  €^€2, 
two  circles  over  ^1^2  and  B1B2  as  diameters  intersect  at  two 
points  (real  or  imaginary)  Fj^,  F^.  Joining  P^  to  -4^,  ^2  and  B^,  B^, 
the  involution  of  the  tangents  from  F^  to  all  conies  of  the  range 
is  determined,  and  as  two  of  these  pairs,  F^A^^,  F^A^  and  F^B^, 
F-JB^,  are  rectangular,  all  other  pairs  are  rectangular;  i.e., 
PjCi-L-FiCs,  and  the  circle  over  CJJ^.  as  a  diameter  is  coaxiil 
with  the  first  two  circles.^  Constructing  all  circles  from  the 
points  of  which  rectangular  pairs  of  tangents  may  be  drawn  to 
the  conies,  it  follows  from  the  last  remarks  that  all  these  circles 
form  a  coaxial  system.  We  state  these  facts  once  more  in  the 
theorem : 

The  circles  from  whose  points  pairs  of  perpendicular  tangents 
may  be  drawn  to  the  conies  of  a  range,  each  for  each,  form  a  coaxial 
system. 

Ex.  I.  If  through  the  vertices  of  two  angles,  whose  sides 
intersect  each  other  in  the  points  A,  B,  C,  D,  two  parallel  Hues 
are  drawn,  then  the  harmonic  lines  of  each  of  these  parallel  lines 
with  respect  to  the  sides  of  the  corresponding  angles  intersect 
each  other  in  a  point  which  describes  the  conic  of  the  middle 
points  of  all  conies  through  A,  B,  C,  D,  when  the  direction  of 
the  two  parallel  lines  changes. 

§  46.   Products  of  Pencils  and  Ranges  of  Conies. 

I.  The  pencils  and  ranges  of  conies  may  be  related  to  each 
other  by  requiring  that  two  conies  shall  correspond  to  each  other 
if  their  equations  are  determined  by  one  and  the  same  param- 
eter X.     Thus,   for  a  certain  value  of  X, 

j  w+M  =0, 
<^'^  I  v+Xv,  =0 

^  The  student  is  asked  to  make  the  foregoing  construction. 


PENCILS  AND  RANGES  OF  CONICS.  i8l 

represent  two  corresponding  conies.  The  two  pencils  (i)  are 
said  to  be  projective.  Two  corresponding  conies  intersect  each 
other  in  four  points  (including  imaginary  points).  We  obtain 
the  locus  of  all  these  points  by  eliminating  X  from  equations  (i), 
which  gives 

(2)  UV^—U{U  =  0, 

an  equation  of  the  fourth  degree  in  x  and  y.     Hence  the  theorem: 

Two  projective  pencils  of  conies  produce  a  curve  of  the  fourth 
order. 

As  the  equation  of  a  curve  of  the  fourth  order  depends  upon 
twelve  constants  and  as  (2)  contains  twenty  constants,  it  is 
evidently  always  possible  to  state  the  converse;   i.e., 

Every  curve  of  the  fourth  order  may  be  considered  as  the  product 
of  two  projective  pencils  of  conies. 

The  curve  as  represented  by  (2)  passes  through  the  inter- 
sections of  U  and  U^,  V  and  Fj,  U  and  V,  U^  and  V^. 

Reciprocally  we  have  the  theorems: 

Two  projective  ranges  of  conies  produce  a  curve  of  the  fourth 
class. 

Every  curve  of  the  fourth  class  may  he  considered  as  the  product 
of  two  projective  ranges  of  conies. 

2.  In  analogy  to  (i)  a  pencil  of  conies  and  a  pencil  of  rays 
are  projective  if  their  equations  may  be  written  in  the  respective 
forms 

\  u+Xu,  =0, 
^^^  }l+K-o. 

For  every  X  we  have  a  conic  and  a  ray  corresponding  to  each 
other  in  this  projectivity,  and  the  two  intersect  each  other  in 
two  points.  The  locus  of  these  points  is  obtained  by  eliminating 
X  from  equations  (3);    so  that  its  equation  is 

(4)  ul^—u-J,  =  o, 


1 82  PROJECTIVE   GEOMETRY. 

and  is  of  the  third  degree.  It  is  satisfied  for  /  =  o,  /i  =  o;  u=o, 
u^  =  o;    u  =  o,  l  =  o;    Wi  =  o,  /i  =  o.     Hence  the  theorem: 

The  product  of  a  pencil  of  conies  and  a  projective  pencil  of 
rays  is  a  curve  of  the  third  order  which  passes  through  the  vertex 
of  the  pencil  of  rays  and  through  the  four  fundamental  points  of 
the  pencil  of  conies. 

As  the  equation  of  a  cubic  depends  upon  nine  constants  and 
as  (4)  contains  fourteen  constants,  it  is  always  possible  to  write 
the  equation  of  any  cubic  in  the  form  of  (4).     Hence  the  theorem: 

Every  cubic  may  be  considered  as  the  product  of  a  pencil  of 
conies  and  a  projective  pencil  of  rays. 

Reciprocally : 

The  product  of  a  range  of  conies  and  a  range  of  points  is  a 
curve  of  the  third  class  which  is  inscribed  to  the  fundamental 
quadrilateral  of  the  range  of  conies  and  which  touches  the  range 
of  points. 

Conversely,  every  curve  of  the  third  class  may  be  considered 
as  such  a  product. 

3.  In  §  45,  3,  it  was  shown  that  the  polars  of  a  point  P  with 
respect  to  a  pencil  of  conies  w4-AWi  =  o  are  concurrent  at  a 
point  P',  and  that  when  P  describes  a  straight  hne,  P'  describes 
a  conic.  In  general  to  a  point  P  corresponds  one  and  only  one 
point  P'.  Let  the  straight  line  described  by  P  be  ^  and  the 
corresponding  conic  described  by  P'  be  G,  and  designate  the 
points  where  g  cuts  G  by  X  and  X',  Fig.  75.  The  relation  between 
P  and  P'  is  involutoric;  i.e.,  all  polars  of  P'  with  respect  to  the 
pencil  pass  through  P. 

To  the  point  X  on  g  corresponds  a  point  on  G,  to  X  on  G  corre- 
sponds a  point  on  g;  but  to  X  only  one  point  corresponds  in  the 
correspondence  between  P  and  P';  hence  the  point  correspond- 
ing to  X  is  X'.  Conversely,  to  X'  corresponds  X.  The  pencil  of 
conies  cuts  g  in  an  involution  of  points.  Let  M  and  N  be  the 
double-points  of  this  involution,  Vm  and  F^  the  conies  of  the 
pencil  touching  ^  at  if  and  N.  Then,  the  polars  of  M  with 
respect  to  Vm  and  V^  are  g  and  the  polar  passing  through  N. 
Hence,  in  the  correspondence  of  P  and  P',  to  M  corresponds 


PENCILS  AND   RANGES   OF   CONICS.  183 

the  point  N,  and,  conversely,  to  N  corresponds  M.  There  are 
only  two  points  on,  g  with  this  property,  the  points  X  and  X'  where 
g  cuts  the  conic  G.  Consequently  M  and  N  coincide  with  X 
and  X'.     We  may  state  this  result  in  the  theorem: 


Fig.  75. 

In  the  correspondence  of  P  and  P'  to  a  straight  line  g  corre- 
sponds a  conic  G.  The  points  on  g  whose  corresponding  points 
are  on  g  itself  are  the  points  of  intersection  X  and  X'  of  g  with  G. 
These  same  points  are  also  the  double-points  of  the  involution  of 
points  which  the  pencil  of  conies  cuts  out  on  g. 

According  to  the*  theorem  that  G  is  also  the  locus  of  the  poles 
of  g  with  respect  to  the  conies  of  the  pencil,  the  points  X  and  X' 
on  G  are  poles  of  g,  and  as  these  coincide  with  g  it  follows  that 
g  touches  two  conies  of  the  pencil  at  X  and  X' ;  in  other  words, 
X  and  X^  are  the  double-points  of  the  involution  cut  out  on  g 
by  the  pencil  of  conies,  as  has  been  established  above.  The 
theorem  therefore  also  holds  for  an  imaginary  pair  of  corre- 
sponding points  X,  X\ 

4.  Consider  now  the  straight  lines  of  a  pencil : 

(5)  {oc+i^0L^)x+(l^+fi^i)y+r+Fn=O' 


i84 


PROJECTIVE  GEOMETRY. 


For  a  definite  value  p.  we  have  a  definite  ray  of  the  pencil. 
According  to  (8)  in  §  45,  3,  when  P  describes  the  .line  (5),  P' 
describes  the  conic 


(6) 


ax^  +  by/  +  d       hx-l  +  cy(  +  e      dx(  +  eyl  +  / 

a^xl  +  h^yl  +  d^     b^x/  +  c^y/  +  e^     d^x^'  +  ^i^'/  +  /i 

a+Zitti  /?+/^/5]  r+/"ri 


which  may  also  be  written  in  the  form 
(7) 


ax^  -\-by^'  +  d        bx^A-cyl-^e        dx/+ey^-\-f 
a^x/  +  b^y/  +  di     b^x/  +  c^y/  +  Ci     f/^^/  +  e^y/  +  /^ 

a  ^  r 


+  ^ 


ax/+by/  +  d        bx^'+cyZ+e      dx/+ey^'  +  f 
a^xl  +  b^yl  +  c^i     b^xl  +  c^}'/  +  g^     d^xl  +  g^)^/  +  /^ 


=0. 


Designating  ax-^^y-\-  j  and  aiX+/5i)'+  ^-^  by  g  and  gi  and  the 
corresponding  conies  by  G  =  o  and  Gi  =  o,  then  to  the  pencil 
g-V  pgx  =  o  corresponds  the  projective  pencil  of  conies  G-\-  iiG^  =  o. 
The  product  of  the  two  pencils  is  therefore  a  curve  of  the  third 
order  with  the  equation 


(8) 


Gg^~G^g  =  o. 


In  the  transformation  of  P  into  P',  to  a  pencil  of  rays  corre- 
sponds a  pencil  of  conies  projective  to  the  ^pencil  of  rays.  The 
product  of  the  two  pencils  is  a  curve  of  the  third  order.  This 
•curve  may  also  be  considered  as  the  locus  of  those  points  on  the 
rays  of  a  pencil  whose  corresponding  points  are  on  the  same  rays, 
each  for  each. 

Ex.  I.  Establish  the  equation  of  a  coaxial  system  of  circles. 
Prove  the  propositions  of  this  section  directly  in  this  special  case. 

Ex.  2.  Prove  that  the  pencil  of  rays  joining  any  point  to 
the  centers  of  a  coaxial  system  of  circles  is  projective  to  this  sys- 
tem. Establish  the  equation  of  the  curve  produced  by  the  two 
pencils. 


THE  STEIN  BRIAN   TRANSFORMATION.  185 

Ex.  3.  Show  in  what  manner  a  system  of  confocal  conies 
may  be  considered  as  a  range  of  conies. 

Ex.  4.  What  is  the  fundamental  quadrilateral  in  case  of  two 
conies  u  =  o,  u^  =  o,  having  a  double  contact? 


§  47.   The  Steinerian  Transformation. 

I.  In  the  foregoing  sections  we  have  shown  that  the  polars 
of  any  point  P  with  respect  to  a  pencil  of  conies  are  concurrent 
at  a  point  P'.  For  the  construction  and  clear  understanding  of 
this  transformation  it  is  of  great  advantage  to  consider  in  par- 
ticular the  degenerate  conies  of  the  pencil  through  the  quadrangle 
which  shall  be  designated  hy  A  ^A^A^A^,  and  its  diagonal  points  by 
B^,B2,Bs.  The  pairs  of  lines  A ,A 2,  B^A 3 ;  A ^A 3,  B,A ^■,A^A^,B^A^ 
are  the  degenerate  conies  of  the  pencil.  To  find  P'  when  P  is  given, 
join  P  to  ^1,  i^j,  B^,  Fig.  76,  and  construct  the  fourth  harmonic 
rays  to  PB^,  PB^,  PB^  with  respect  to  the  corresponding  pairs 
of  lines  through  B^,  B^,  B3.  The  three  harmonic  rays  intersect 
each  other  at  P\  From  this  simple  geometric  construction  it 
is  now  easy  to  study  the  correspondence  of  P  and  P'  for  any 
particular  positions.  At  every  point  B,  say  B3,  the  lines  A^A^, 
B3A3,  B^Bi,  B^B^  form  a  harmonic  pencil.  To  the  points  B 
correspond,  therefore,  all  points  of  their  opposite  sides  of  the  tri- 
angle B1B2B3.  The  points  ^i,^2>^3>^4  are  invariant,  since  the 
fourth  harmonic  rays  pass  through  the  points  themselves.  To 
a  point  on  any  hne  joining  two  of  the  fundamental  points,  say 
A^A^,  corresponds  the  fourth  harmonic  point  to  the  pair  A^A^. 
All  other  points  are  in  uniform  correspondence. 

We  have  seen  that  to  a  straight  Hne  corresponds  a  conic.  As 
a  straight  line  cuts  each  of  the  sides  B^B^,  B^B^,  BJS^,  and  as  to 
these  sides  correspond  the  opposite  points  B^,  B^,  B2,  it  follows 
that  said  conic  passes  through  the  points  B^,  B^,  B3.  To  the 
straight  lines  of  the  plane  corresponds  the  net  of  conies  through 

1  See  Steiner's  collected  works,  Vol.  I,  pp.  407-421,  and  M.  Disteli:  Die 
Metrik  der  circularen  Curven  dritter  Ordnung  im  Zusammenhang  mit  geometrischen 
Lehrsdtzen  Jakob  Steiners.    Also  Poncelet:  Traite,  i  ed.  1822,  p.  198. 


i86 


PROJECTIVE  GEOMETRY. 


B^B^Bs.  Taking  a  pencil  of  rays  through  P,  its  corresponding 
pencil  of  conies  passes  through  P'  and  Bj^,  B^,  B^.  The  curve 
of  the  third  order  produced  by  these  two  pencils  passes,  therefore, 
according  to  §  46,  (4),  through  P  and  P' ,  B^,  B^,  B^. 

On  every  ray  through  P  there  are  two  corresponding  points 
X  and  X'  of  the  cubic.     Consequently,  connecting  P  to  ^  1,  ^  2> 


Fig.  76. 

As,  A^,  the  corresponding  points  on  these  four  rays  coincide, 
each  for  each,  with  A^,  A  2,  A^,  A^,  so  that  these  points  are  on 
the  cubic  and  PA^,  PA^,PA^,  PA^  the  tangents  at  these  points. 
On  the  rays  PB^,  PB2,  PB^  the  points  which  correspond  to  B^,  B^, 
B3  are  the  points  of  intersection  5/,  B2,  B^  of  these  rays  with  the 
sides  ^2-^35  BaBi,  B^B^,  respectively.  The  points  B^',  B^,  B^  are 
therefore  also  on  the  cubic.     Hence  the  theorem: 


THE  STEIN  BRIAN    TRANSFORMATION. 


187 


In  the  Steinerian  transformation  to  every  pencil  0}  rays  corre- 
sponds a  projective  pencil  of  conies  through  the  diagonal  points 
of  the  fundamental  quadrilateral.  The  product  of  the  two  pencils 
is  a  curve  of  the  third  order  through  the  vertex  of  the  pencil  of 
rays  and  its  corresponding  point  and  through  the  vertices  and  diag- 
onal points  of  the  fundamental  quadrangle.  Thus  to  every  point  of 
the  plane  may  he  associated  a  certain  curve  of  the  third  order  in 
the  Steinerian  transformation.  All  these  00 ^  cubics  pass  through 
seven  fixed  points. 

Without  proceeding  to  the  Steinerian  transformation  of 
conies,  cubics,  etc.,  we  shall  immediately  take  the  general  case 
of  a  curve  of  the  nth.  order,  Cn-  To  determine  in  how  many  points 
any  straight  line  g  cuts  Cn,  notice  that  the  conic  G  corresponding 
to  g  cuts  Cn  in  2W  points.  Hence,  when  the  whole  configura- 
tion is  transformed,  G  with  its  2W  intersections  on  Cn  is  trans- 
formed into  2W  intersections  of  g  with  the  transformed  Cn-  Hence 
the  theorem: 

In  a  Steinerian  transformation  a  curve  of  the  nth  order  is  gen- 
erally transformed  into  a  curve  of  order  2n. 

2.  Analytical  Expression  for  a  Steinerian  Transformation. 

Nothing  will  be  lost  in  the  general  result  if  we  assume  that 
the  points  A^,  A 2,  A3  form  an  equi-    A^ 
lateral  triangle  and  that  ^4  be  its 
center,   since  by  a  collineation  this 
orthogonal  quadrangle  may  be  trans- 
formed into  any  other  quadrangle. 
Let  A^  coincide  with  the  origin,  andg  . 
A I   with    the   X-axis,   Fig.    77,   and 
AiAi=A2A^=AsAi=i.     The    foot- 
points  of  the  perpendiculars  of  the 
triangle   are   Bj^,   B2,   B3.     Consider 
first  the  degenerate  conic  consisting 
of  the  lines  A^A2  and  A^B^  with  the 
equations 


Fig.  77. 


1=0, 


l88  PROJECTIVE   GEOMETRY. 

SO  that  the  equation  of  the  degenerate  conic  is 

(1)  \^•^^-^^rj'^+2^rj-^y^■^+■f)  =  0. 

Similarly  the  equation  of  the  degenerate  conic  represented  by 
^1^1 3  and  ^42^2  is 

(2)  \/2)-^^-^-'r}'^-2^y-^2>-^-f}  =  o. 

The  equations  of  the  polars  of  the  point  P  (x,  y)  with  respect 
to  these  conies  are 


(3)  (W3  +  )'-|V3)e+(.T-3;\/3  +  i)7;-^\/3  +  ^=o, 


(4)  {oc^/  1,-y-^^y ^)^-  {x+ys/ ^  +  \)ri--^/ ^-^^  =  0. 

The  common  solutions  of  (3)  and  (4)  are  the  coordinates  a/,  y' 
of  the  point  P  corresponding  to  P  in  the  Steinerian  transformation: 

f     ,     2{x'^—y'^)-\-x 

Solving  these,  equations  with  respect  to  x  and  y  we  obtain 

f         2{x'^-y'^)  +  x' 

(6)    ■-- ^%(^;+/p-.'       . 

y  —4fXy 


y- 


4(^2^  y  2^  _  J  . 


which  shows  that  the  transformation  is  involutoric. 

y 
To  the  line  at  infinity,  :r=oo,  ;y=co,  —  =  arbitr.,  corresponds 

the  circle 

(7)  A/2+y^=i. 


CURVES  OF   THE    THIRD   ORDER.  1 89 

Ex.  I.  The  centers  of  the  conies  circumscribed  to  a  quad- 
rangle A-i^A^A^A^  he  on  a  conic  K,  which  bisects  the  distances 
between  these  points  in  six  points.  These  form  three  parallelo- 
grams having  the  same  center,  which  is  the  center  of  the  conic 
cutting  out  these  points. 

Ex.  2.  According  as  a  straight  g  cuts  K,  in  two  real  or  two 
imaginary  points,  or  touches  it,  the  corresponding  conic  in  the 
Steinerian  transformation  will  be  an  hyperbola,  an  ellipse,  or 
a  parabola. 

Ex.  3.  Prove  by  formulas  (5)  that  in  a  Steinerian  transforma- 
tion a  Cn  is  transformed  into  a  Cjn-  In  particular  a  straight  line 
is  transformed  into  a  conic. 

Ex.  4.  Prove  that  in  the  Steinerian  transformation  A^A^A^A^ 
are  invariant  points  and  that  to  the  sides  B^B.^,  B^B^,  B^B^  corre- 
spond the  points  B3,  B^,  B^,  by  using  formulas  (5). 

§  48.    Curves  of  the  Third  Order. 

I.  In  the  Steinerian  transformation,  with  every  point  of  the 
plane  is  associated  a  certain  cubic.  As  in  the  previous  section 
assume  as  conies  determining  the  fundamental  quadrangle  or 
quadruple  the  degenerate  conies. 

(i)  u  ='\/2,-x'^-\-2xy—\^2)'y'^~'^  Z'^'^y^^i 

(2)  u^  =  \/i-x'^—2xy—\/7^-y'^—\/2>'^~y""'^- 

To  find  the  cubic  associated  with  the  point  {x' ,  y'),  take,  as 
lines  g  and  g^  in  formula  8,  §  46, 

(3)  g  =  x-x'  =  o, 

(4)  gi=y-y'=o. 

According  to  (5),  §  47,  to  these  lines  correspond  in  the  Stei- 
nerian transformation  the  conies 

(5)  G^2(x2-/)  +  x-x'|4(x2+/)-i|=o, 

(6)  Gi=      y-/^xy      -/|4(x2+/)-i}=o. 


igo  PROJECTIVE  GEOMETRY. 

The  equation  of  the  cubic  associated  with  the  point  (x' ,  y') 
is  G^i-Gi^  =  o,  or 

(y— y')  { 2  (^^—  y^) + ^—  oc'[^(x'^+y'^)  —  i]  | 

-(:r-x')|:y-4X3'  -/[4(^'+)'')-i]}=o, 


or 


(7) 


(:y-/) 


4(^2  + /)■ 


i-^) 


r^""*'H4(x"+/)-i"^'=°- 


From  the  form  of  this  equation  it  is  apparent  that  a  Steinerian 
transformation  does  not  change  the  equation.     Hence  the  theorem ; 

The  net  of  -cubics  through  a  quadruple  and  its  diagonal  points 
is  invaria7it  in  the  corresponding  Steinerian  transformation. 

This  is  also  geometrically  evident.  In  the  construction  of 
the  curve,  Fig.  76,  eleven  points  are  obtained  through  which 
the  cubic  passes  and  which,  as  a  group,  are  invariant  in  the 
Steinerian  transformation. 

y' 

For  the  points  x'=oo,  y=oo,  -j^k-,  (7)  reduces  to 


y— /[xy  2(x^— 7^)  +  ^ 

(8)  y+  '^^+~rv^ — 2^^ '^~r^~^ — 2^ — =o- 


Also  in  this  case  the  cubic  is  the  locus  of  the  double-points 
of  the  involutions  cut  out  on  the  pencil  of  parallel  rays  through 

the  infinite  point  [—,=  k\  by  the    pencil  of  conies  through  the 

fundamental  quadruple.  The  line  at  infinity  belongs  also  to  the 
pencil  of  parallel  rays,  and  as  the  involution  on  it  is  rectangular 
it  follows  that  the  double  points  are  the  circular  points.  Hence 
(8)  represents  a  pencil  of  bicircular  cubics. 

As  has  been  seen  already,  the  tangents  to  the  cubic  at  the  points 
A^A^A^A^  pass  through  the  point  P.  We  shall  now  prove  that 
the  tangents  at  B^,  B2,  B3  pass  through  P'.     For  this  purpose 


CURVES  OF    THE    THIRD   ORDER. 


191 


draw  a  ray  through  P  cutting  the  cubic  in  two  points  U  and  F, 
of  which  U  shall  be  close  to  B2,  Fig.  78.     To  this  ray  corresponds 


Fig.  78. 


in  the  Steinerian  transformation  a  conic  through  B^,  B2,  B3,  U, 
V,  and  P'.  As  the  ray  through  P  turns  in  such  a  manner  that  U 
approaches  B2  as  a  hmit,  the  corresponding  conic  will  approach 
the  degenerated  conic,  consisting  of  the  ray  P'^2  and  the  side 
B^Bs  as  a  limit.  Hence,  when  the  ray  passes  through  B^,  the 
corresponding  ray  through  P'  will  be  a  tangent  to  the  cubic  at  B^. 
A  similar  result  is  obtained  for  the  points  B^  and  B2,  which  proves 
the  proposition. 


192 


PROJECTIVE  GEOMETRY. 


2.  In  what  follows  it  will  be  assumed  that  the  cubic  is  a 
circular  curve;  i.e.,  that  the  point  P  is  infinitely  distant.  Desig- 
nating this  infinitely  distant 
point  by  B  and  its  correspond- 
ing point  by  C,  the  tangents  at 
the  A 's  are  parallel  to  the  direc- 
tion of  B,  and  the  tangents  at 
the  5's  pass  through  C,  Fig.  79. 
The  ray  through  C  parallel  to 
the  direction  of  B  is  the  asymp- 
tote of  the  curve.  Hence  the 
tangents  at  the  points  B,  B^,  B2, 
Bs  meet  in  the  point  C  of  the 
same  curve.  Four  points  on 
the  cubic  with  this  property 
are  called  a  Steinerian  quad- 
ruple of  the  cubic. 

Thus  A^A^A^A,,  BB^B^B^ 
are  such  quadruples.  Accord- 
ing to  previous  results,  the  rays 
BB^,  BB2,  BB^  cut  the  opposite 
sides  ^2-^3)  ^3^ij  -^1-^2  ill  three 
more  points,  C^,  C2,  Cg,  of  the 
cubic.  But  this  is  equivalent 
with  considering  BB1B2B3  as  a 
fundamental  quadrangle  in  a 
new  Steinerian  transformation 
with  Ci,  C2,  C3  as  the  diagonal 
points,  and  C  as  the  original 
point  associated  with  the  cubic. 
That  the  cubic  associated  with 
C  in  this  new  transformation  is 
identical  with  the  original  cubic 
follows  from  the  following  consideration:  The  points  B  beuig 
points  of  tangency  count  for  eight  given  points.  Furthermore, 
the  four  C's  lie  on  the  original  curve,  so  that  the  new  curve  has 


Fig.  79. 


CURVES   OF   THE    THIRD   ORDER.  193 

at  least  twelve  points  in  common  with  the  original  cubic,  and  is 
consequently  identical  with  it. 

In  this  new  Steinerian  transformation  construct  the  point  D 
corresponding  to  C.  Then  take  the  new  quadruple  CC1C2CS  and 
construct  the  associated  cubic  in  the  Steinerian  transformation 
belonging  to  this  quadruple.  The  new  cubic  is  identical  with 
the  original  cubic,  as  can  easily  be  proved.  The  tangents  at 
C,  Ci,  C2,  C3  all  pass  through  D.  For  the  quadruple  CC^C^C^ 
construct  the  diagonal  points  D^,  D^,  D^.  These  together  with  D 
form  a  new  quadruple,  whose  tangents  pass  through  E,  the  point 
corresponding  to  D  in  the  transformation  associated  with  the 
quadruple  C^C^C^C^.  Continuing  this  construction,^  we  may 
obtain  any  number  of  points  of  the  cubic  arranged  in  quadruples. 
The  points  B,  C,  D,  E  .  .  .  have  the  property  that  the  tangent  at 
one  of  these  points  always  passes  through  the  previous  point. 

3.  The  general  equation  of  a  cubic  may  be  written 

(i)  Ax^+Bx-y+Cxy'+Dx^^ 

ax"^ -\- 2hxy -\- cy^ -\-  2dx-\-2ey-{-j  =  o. 

The  problem  arises,  what  connection  exists  between  the 
fundamental  quadruple  with  which  the  cubic  is  associated  and 
the  shape  or  equation  of  the  cubic.  In  the  above  discussion  the 
quadruple  was  assumed  as  real  and  the  cubic  consisted  of  a 
serpentine  (infinite  branch)  and  an  oval.  By  certain  collineations 
this  curve  may  be  transformed  into  various  other  curves  which 
may  be  characterized  with  respect  to  their  behavior  at  infinity. 
The  serpentine  or  oval  will  be  called  elliptic,  h)rperbolic,  or  para- 
bolic, according  as  they  have  two  imaginary,  two  real,  or  two 
coincident  points  at  infinity.  Designating  by  r  the  counter-axis 
which  in  a  coUineation  is  transformed  to  infinity,  and  by  S  and  O 
the  serpentine  and  oval  of  the  cubic,  then  the  transformed  curves 
resulting  from  various  positions  of  r  are  as  given  in  the  following 
table : 

^  For  the  sake  of  simplicity,  in  the  figure  only  the  quadruple  CCfij^z  ^'^^  been 
constructed. 


194 


PROJECTIVE   GEOMETRY. 


r 

Original  Curve. 

Resulting  Curve. 

S 

0 

5 

0 

cutting 
tangent 
cutting 
tangent 

in  3  points 
in  I  point 

hyperbolic 
parabolic 
elUptic 
elliptic 

elhptic 
elliptic 
hyperbolic 
parabolic 

in  2  points 
in  I  point 

From  these  possible  coUineations  it  is  seen  that  a  cubic  with 
two  branches,  serpentine  and  oval,  by  any  coUineation  is  trans- 
formed into  a  cubic  with  two  branches.  The  geometrical  dis- 
cussion of  this  section  therefore  does  not  cover  all  cases  as  repre- 
sented by  the  general  equation  of  the  cubic.  For  this  purpose 
it  is  necessary  to  classify  the  cubics  from  the  general  equation, 
or  the  fundamental  quadruple,  by  introducing  coincident  and 
imaginary  elements.  We  shall  do  both.  As  the  analytical  dis- 
cussion is  briefer,  we  shall  take  this  up  first  and  discuss  the  geomet- 
rical aspect  later  on.  To  equation  (i)  apply  the  general  projec- 
tive transformation  or  collineation  of  the  x}'-plane  as  given  in 
§  19.  This  collineation  depends  upon  eight  parameters.  After 
the  transformation,  clearing  of  fractions,  collection  of  equal  terms 
in  X  and  y,  (i)  assumes  the  form 


(2) 


aiX^+  2byXy^Cyy''-^  2d^x^  2e^y-\-  /i  =  0, 


where  yl^,  B^, .  .  .  a^,  b^^,  .  .  .  are  polynomials  in  A,B,  ...  a,b,  .  .  . 
and  the  eight  parameters  of  the  collineation.  It  is  evidently 
possible  to  choose  in  an  infinite  number  of  ways  the  eight  param- 
eters in  such  a  manner  that  in  (2)  the  coefficients  Bj^,  Cj,  D^, 
&i,  Ci  vanish,  which  amounts  to  five  equations  with  eight  un- 
known quantities.  It  is  therefore  possible  to  find  a  collineation 
transforming  (i)  into  an  equation  of  the  form 


;y^=ax^+/?x^-|-  yx-^  d, 


CURVES   OF   THE   THIRD   ORDER. 


195 


or,  resolving  the  right  side  into  its  hnear  factors, 


(3) 


y^=a(x-  e^)  (x-  e^  {x-  e^ , 


in  which  e^  has  a  different  meaning  from  the  e^  used  above. 

The  general  equation  of  the  cubic  can  therefore  always  be 
reduced  to  an  equation  of  the  form  (3),  so  that  the  discussion  of 
the  cubic  with  respect  to  its  type  may  be  limited  to  equation  (3). 
This  equation  represents  a  curv^e 
which  is  symmetrical  with  respect 
to  the  X-axis,  and  its  shape  depends 
essentially  upon  the  values  of  e^, 
62,  €3.  Assume  e^^e^^e^.  The 
following  cases  may  be  dis- 
tinguished : 

I.  gj,  62,  ^3  are  real  and  different 
from  each  other. 

On  the  X-axis  the  curve  has 
the  real  points  with  the  abscissas 
e„  e2,  e^,  Fig.  80. 


Fig.  80. 


-"+ ' 


In  order  that  y^  be  positive,  it  is  necessary  that 
either  e^^x<e2  or  x^e^.  From  this  it 
follows  easily  that  the  cubic  consists  in 
this  case  of  an  oval  and  a  serpentine. 
This  is  the  case  discussed  in  connection 
with  the  real  quadruple. 

II.  ^1  is  real,  gj  ^^^  ^3  0.^^  conjugate 
imaginary.  / 

In   this   case  we  can  write  (3)  in  the 
form 

y'^  =  a{x-e,)[{x-py  +  g^l 
from  which  follows  that  y^  is  real  only 
whenx>ei;  the  curve  consists  of  only 
one  branch.  Fig.  81.  This  case  is  equivalent  with  a  fundamental 
quadruple  with  two  real  and  two  conjugate  imaginary  points,  as 
we  shall  see  later  on. 


Fig.  81. 


igfr 


PROJECTIVE  GEOMETRY. 


9\. _V^._. 


Fig.  82, 


III.  ^1  =  ^2  ^wJ  63  all  real. 
Equation  (3)  assumes  the  form 

To  get  real  values  for  y,  x>fs- 
The  point  x  =  e^,  y  =  o  satisfies  the 
equation  also;  but  it  is  an  isolated 
point,  Fig.  82. 

Correspondingly,  in  the  quadruple 
two  points  are  real  and  two  coincide. 

IV.  e^  and  62  =  e^  are  real. 
The  equation  becomes 

y^=a(x—ei){x—  e^)  ^ 

y  is   real  for  x^e^.     Hence  x  =  e2  is   a 
double-point  of  the  cubic.  Fig.  83. 

This    case    corresponds   to   a   funda- 
mental  quadruple    with   two    coincident    ' 
and  two  conjugate  imaginary  vertices. 

Equation  (3)  can  be  written 

We  must  take  x^e^.  The  curve 
has  a  cusp  at  x  =  ei  with  the  x-axis 
as  a  tangent,  Fig.  84.     The  four  points  of  the  quadruple  are 

real  and  three  of  them  are  coinci- 
dent along  the  tangent  of  the  cusp. 
These  are  the  five  types  of 
curves  of  the  third  order  into 
which  all  cubics  may  be  projected. 
Newton  ^  called  these  five  types, 
found  by  him,  respectively, 

parabola     campaniformis     cum 
ovali, 

parabola  pura, 
parabola  puncta, 


Fig.  83. 


Fig.  84. 


Enmneratio  lineartim  tertii  ordinis  (Londini,  1706). 


CURVES  OF  THE   THIRD   ORDER.  197 

parabola  nodata, 

parabola  cuspidata, 
and,   according  to  their  behavior  at  infinity,   subdivided  them 
into  seventy-two  different  kinds.     By  later  investigations  six  more 
were  added  to  the  seventy-two. 

As  in  the  first  case,  this  classification  may  be  made  by  choos- 
ing in  the  perspective  colHneations  the  counter-axes  r  properly. 

§  49.   Curves    of    the    Third    Order    Generated    by    Involutoric 

Pencils. 

I.  Every  straight  fine  cuts  a  pencil  of  conies  in  an  involution 
of  points.  Instead  of  any  two  conies  ■w=o,  %  =  o,  we  may  take 
two  degenerate  conies  with  the  same  vertex: 

where  p  and  p^  represent  two  distinct  straight  lines.  The  pencil 
of  conies  then  becomes 

u+vu,^pp,+  v\{p+Xp,){p+  PlP,)\=o, 

where  v  is  a  variable  parameter.  We  may  write  this  also  in  the 
form 

vp-'+^i  -h  VIX+  vX)pp,+  vXixp^^^o. 

Solving  for  p, 

^     -{i  +  vX+  vji)  ±  \/(i  +  vX+  vpty-  ^v^fi 
P=  ^^ ^ -A, 

and  designating  by  $  and  rj  the  expressions  multiplying  p^  in  the 
last  formula,  the  equation  of  the  involutoric  pencil  of  rays  may 
be  written 

(i)  (p-^Px){p-Vpi)=o, 

where  I'tj==^/z  =  constant. 


198  PROJECTIVE   GEOMETRY. 

For  every  set  of  values  of  f  and  tj  satisfying  this  condition,  (i) 
represents  two  rays  of  a  pair  of  the  involution 

p--f]pi  =  o. 

The  product  of  two  projective  pencils  of  this  kind,  having 
the  same  v, 

(2)  M+  W  {p+  ^Pi)iP+  {J-Px)  \  =0, 

(3)  ??i+  ^i  (?+  ^'?i)(?+/?i)  1  =0, 
is  evidently  the  curve  of  the  fourth  order: 

(4)  PPx  I  iq+  ^'(ix)  (?+  /?.i)  1  -  ??i  l(i'+  ^A)  CP+  M)  \  =0, 

with  the  double-points  p=pi  =  o  and  q  =  qi  =  o. 

In  (1),  p  =  o  and  j>i=o  are  evidently  the  equations  of  a  pair  of 
the  involution. 

In  (3)  the  corresponding  pair  is  given  by  q=o,  qi=o.  Letting 
the  corresponding  rays  p^  and  q^  coincide;  i.e.,  pi  =  qi  =  o,  the 
curve  (4)  degenerates  immediately  into  the  ray  pi=o  and  the 
cubic 

(5)  p{(q+^'Pi)  (q+}i'Pi)]-q\(P+^Pi)  (i'+M)}=o. 

To  distinguish  the  pencils  (2)  and  (3)  from  ordinary  linear 
involutoric  pencils,  we  shall  call  them  quadratic.  The  result 
may  be  stated  in  the  theorem: 

The  product  0}  two  projective  quadratic  involutions  of  rays 
is  a  curve  0}  the  jourth  order.  If  the  two  involutions  have  two 
corresponding  rays  in  common,  then  their  product  is  a  curve  of 
the  third  order  and  that  common  ray. 

The  cubic  can  also  be  produced  by  two  projective  pencils  of 
which  one  is  linear  and  one  quadratic: 

(6)  I   ^+^^" 

^  ^  [qqi+Aiq+ki)  (q+mi)}> 


CURVES  OF    THE   THIRD   ORDER.  '  1 99 

whose  product  is 

(7)  P\(q+  ^?i)  (?+  Ml)  I  -  A??i = o- 

As  (5)  and  (7)  contain  respectively  twelve  and  ten  arbitrary 
parameters  it  is  clear  that  every  cubic  may  be  represented  by 
one  of  these  forms. 

2.  Considering  (5),  each  two  pairs  of  corresponding  rays  of 
the  two  quadratic  involutions  (2)  and  (3),  in  which  qt  =  pi, 

(8)  ■  j^-fft-o, 

'5^  }i-r/P.  =  o, 

intersect  each  other  in  four  points  of  the  cubic.  The  vertices 
of  the  two  pencils  are  also  on  the  cubic.  Two  pairs  of  the  quad- 
ratic involution  in  one  pencil  and  the  two  corresponding 
pairs  in  the  other  pencil  are  therefore  sufficient  to  determine 
the  projectivity  and  consequently  also  the  cubic,  since  they  deter- 
mine ten  points  on  the  curve. 

Conversely,  if  on  a  cubic  two  vertices  B  and  B^  are  known, 
and  if  it  occurs  twice  that  two  rays  through  B  cut  certain  two  rays 
through  Bi  in  four  points  of  the  cubic,  then  these  pairs  determine 
two  projective  quadratic  involutions  of  rays  by  which  the  entire 
cubic  is  produced. 

To  prove  this  assume  a  ray,  a,  through  B^  passing  very  close 
to  B.  If  the  foregoing  statement  would  not  be  true,  the 
product  of  the  involutions  determined  by  the  four  pairs 
of  rays  through  B  and  Bj^  would  be  a  curve  of  the  fourth  order, 
according  to  (4).  On  the  ray  a  there  would  be  two  points 
cut  out  by  the  corresponding  rays  through  B,  which  in  general 
would  be  distinct.  As  the  ray  a  in  the  limit  approaches  the 
ray  through  B^  passing  through  B,  these  two  points  on  a  become 
coincident;  i.e.,  5  is  a  double-point  of  the  curve  (4).  Similarly 
B^  is    also  a  double-point.      A  double-point  on  another  curve 


200  PROJECTIVE  GEOMETRY. 

counts  for  two  points  of  intersection,  so  that  the  supposed  curve 
of  the  fourth  order  has  twelve  points  in  common  with  the  given 
cubic.  Construct  the  net  of  quartics  (curves  of  the  fourth  order) 
through  these  twelve  points.  Any  two  points  different  from  these 
twelve  points,  with  these,  determine  fourteen  points;  i.e.,  such 
a  quartic  and  only  one,  which  therefore  consists  of  the  given  cubic 
and  the  straight  line  through  the  assumed  two  points. 

From  this  it  follows  that  there  is  only  one  quartic  through 
the  twelve  points  having  B  and  B^  as  double-points,  and  this 
consists  of  the  given  cubic  and  the  line  through  B  and  B^, 
which  proves  the  proposition. 

3.  Consider  three  quadratic  involutions  of  pencils  of  rays 
projective  to  each  other: 

(10)  M+^i(i'+^A)(/'+M)l  =0, 

(11)  qqi+A{q+-^'(ix){<i+!J-'qi)\  =0, 

(12)  rr,+  v{{r+r'r,){r+fi"r,)]  =o, 

and  suppose  that  the  product  of  (lo)  and  (ii)  is  identical  with 
the  product  of  (lo)  and  (12);  i.e.,  that  the  two  equations 

(13)      PPA{q+  ^'qi)(q+  /?i)  1- 1  (P+  ^Pi)(P+  M)  l??i=o, 

(14)  pp, { ir+  ^'VO (r+  ii'%)  \-{{p+  Xp,) (p+  fip,)  ]rr, = o, 

must  be  simultaneously  satisfied  for  all  sets  of  values  of  x  and  y. 
This  can  only  be  true  if 

(15)  ??il(^+/VO(r+/V,)Pfr,l(?+/?i)(g+/?,)|=o 

simultaneously  with  (13)  and  (14). 

From  this  the  theorem  follows: 

J/  a  quadratic  involution  0}  rays  produces  with  two  projective 
quadratic  involutions  one  and  the  same  quartic  or  cubic,  then  the 
product  oj  the  last  two  involutions  is  the  same  quartic  or  cubic. 


CURVES  OF   THE    THIRD    ORDER.  20i 

4.  Considering  again  the  construction  of  a  cubic  by  the  Stein- 
erian  transformation,  Fig.  85,  and  taking  B  at  an  infinite  distance 


Fig.  85. 

in  the  indicated  direction,  then  to  a  point  X  on  the  cubic  corresponds 
a  point  F  on  a  ray  through  X  parallel  to  this  direction.     Joining 


202  PROJECTIVE  GEOMETRY. 

X  and  Y  with  B^  and  producing  XB^  and  YB^  to  their  intersec- 
tions Fi  and  X^  with  the  cubic,  then  X^  corresponds  to  F^  in  the 
Steinerian  transformation  and  hence  X^Y^  is  parallel  to  XY. 
It  is  now  clear  that  the  pairs  of  rays  XY^,  X^F  through  B2  and 
XY,  XjlFj  through  B,  furthermore  the  pairs  A^A^  (counted 
twice)  through  B^  and  BA^,  BA^,  determine  two  projective  quad- 
ratic involutions  around  B2  and  B  whose  product  is  a  cubic, 
since  they  have  the  ray  B2B  in  common.  This  cubic  having 
ten  points  in  common  with  the  cubic  of  the  Steinerian  trans- 
formation (^1,  As',  each  counted  twice,  since  BAj^,  BA^  are  tan- 
gents at  A 1  and  A^;  B,  B2,  X^  Y,  A"^,  F^)  is  identical  with  it.  If 
we  connect  X  and  F  with  B^  and  produce  XB^  and  YB^  to  their 
intersections  AY  and  F/  with  the  cubic,  then  AY  and  F/  cor- 
respond to  each  other  in  the  Steinerian  transformation;  i.e., 
X/F/IIXF.  Consequently  the  cubic  may  also  be  considered 
as  the  product  of  the  involution  around  B  and  an  involution 
around  ^3.  In  the  same  manner  it  is  also  the  product  of  the 
involutions  around  B  and  B^;  hence,  according  to  the  foregoing 
theorem,  the  cubic  is  also  the  product  of  the  two  involutions 
around  ^g.and  B^.  Hence  the  points  where  X^B^  and  Y^B^  pro- 
duced meet  the  cubic  are  the  same  as  X/,  F/.  In  a  similar 
manner  it  can  be  proved  that  XB^  and  X^B^,  YBj^  and  Y^B^ 
intersect  each  other  in  the  points  X',  Y'  of  the  cubic,  so  that 
X'Y'  II XF.  X  has  been  assumed  as  any  point  of  the  cubic,  and 
A'l  in  such  a  manner  that  the  corresponding  point  F  of  X  lies 
in  a  straight  line  with  X^  and  B^.  Consider  now  the  pairs  of 
rays  XX/,  XB^  and  X^X',  X^B^;  and  XF,  XF^  and  X^F^,  X^F; 
they  determine  two  projective  quadratic  involutions  about  X 
and  X^  whose  product  is  a  cubic  which  is  identical  with  the  original 
cubic,  since  it  has  ten  points  in  common  with  it.  Taking  any 
point  G  on  the  cubic  ^  and  letting  XG  and  A'^G  cut  the  cubic  in 
/  and  K,  then  XJ  and  XK  produced  cut  the  cubic  in  one  and 
the  same  point  H;  XG,  XH  and  X^G,  X^H  form  two  corre- 
sponding pairs  of  the  involutions  around  X  and  X^.  If  G  ap- 
proaches the  point  of  intersection  of  XX^  with  the  cubic,  then  / 

'■  For  the  sake  of  simplicity  in  the  figure  the  following  part  of  the  construction 
is  not  shown. 


CURVES  OF   THE    THIRD    ORDER.  203 

and  K  approach  X^  and  A^ ;  hence,  in  the  Hmit,  the  tangents  to 
the  cubic  at  X  and  A\  intersect  each  other  in  a  point  of  the  cubic. 
In  a  similar  manner  it  can  be  proved  that  the  tangents  at  Y  and 
Fj,  X'  and  A/,  Y'  .and  F/  intersect  each  other  in  points  of  the 
cubic.  Again,  G  and  H,  and  /  and  K  may  be  assumed  as 
vertices  of  projective  quadratic  involutions  producing  the  cubic. 
Hence  also  the  tangents  at  G  and  H,  and  /  and  K  intersect 
each  other  in  points  of  the  cubic.  We  can  therefore  state  the 
following  theorem: 

Designating  two  points  on  a  cubic  whose  tangents  at  those 
points  intersect  each  other  in  a  point  of  the  cubic  as  a  Steinerian 
couple,  or  simply  as  a  couple,  then  the  cubic  can  be  produced  by 
two  projective  quadratic  involutions  around  these  points. 

The  lines  joining  any  point  of  the  cubic  to  the  points  of  a  couple 
cut  the  cubic  again  in  a  couple,  and  all  couples  of  the  cubic  are 
produced  when  this  point  describes  the  whole  cubic. 

Each  two  corresponding  pairs  of  the  involutions  around  the 
two  points  of  a  couple  iyitersect  each  other  hi  two  new  couples. 
Such  two  involutions  produce  all  couples  of  the  cubic. 

A  quadruple  on  a  cubic  is  defined  as  a  group  of  four  points 
any  two  of  which  form  a  couple;  i.e.,  the  tangents  at  the  four 
points  concur  in  a  point  of  the  cubic.  From  this  definition  we 
infer  easily: 

The  lines  joining  four  points  of  a  quadruple  cut  the  cubic  in 
another  quadruple.  The  sixteen  lines  joining  the  points  of  two 
quadruples  intersect  each  other,  four  by  four,  in  four  points  of  a 
new  quadruple.  / 

These  results  form  a  part  of  the  theory  of  problems  of  clo- 
sure on  the  cubic  as  it  has  been  developed  by  Steiner,  Clebsch, 
and  others.^  They  are  sufficient  for  the  applications  in  the 
following  sections. 

^  For  further  details  references  are  made  to 

Clebsch:    Crelle's  Journal,  Vol.  LXIII,   pp.   94-121. 

Steiner:     Crelle's  Journal,  Vol.  XXXII,  pp.  371-373- 

DiSTELi:  Die  Steiner' schen  Schliessungsprohleme  nach  darstellend-geometrischer 
Methode.     Leipzig,  1888. 

Emch:  Applications  of  Elliptic  Functions  to  Problems  of  Closure,  University 
of  Colorado  Studies,  Vol.  I,  pp.  81-133. 


204  PROJECTIVE   GEOMETRY. 

Ex.  I.  Verify  the  theorems  of  this  section  constructively, 
when  B  is  finite  or  infinite. 

Ex.  2.  What  relation  must  exist  between  a  quadratic  and  a 
projective  Hnear  involution  of  rays  in  order  that  the  cubic  pro- 
duced be  one  with  a  cusp. 

Ex.  3.  Prove  directly  that  a  cubic  can  be  produced  by  two 
quadratic  involutions  around  the  points  of  a  couple  by  determin- 
ing two  corresponding  pairs  of  the  involutions. 


§  50.   Various  Methods  of  Generating  a  Circular  Cubic. 

I.  In  §  48  (8),  we  found  for  the  equation  of  the  bicircular 
cubic  referred  to  the  equilateral  triangle  A^A^A^  with  A^  as 
point  of  concurrence  of  altitudes,  after  some  rearrangement, 

(i)         ^{kx—  y)  {x'^+y'^)  —  2  kx^—  A,xy-\-y-\-  ly"^—  2  kx-\-  2y  =  o. 

The  slopes  of  the  asymptotes  at  the  circular  points  are  evi- 
dently +  i  and  —  i,  so  that  the  equations  of  these  asymptotes 
are  of  the  form 

'  y  =  ix+c^, 


(2) 

where  c^  and  c^  are  constants  to  be  determined.  If  equations  (2) 
represent  asymptotes,  then  their  common  solutions  with  (i) 
must  give  infinite  values  for  x  and  y.  Substituting  the  values 
vof  y  from  (2)  in  (i),  w^e  get  respectively 

^^^  I  {^c^-d>c^Ki+/[i-/[K)x'^+B'x+C'=o, 

where  B,  C;  B\  C  are  polynomials  in  c^,  k;  Cj,  k,  different 
from  those  of  the  x^'s. 

In  both  cases  the  values  of  x  will  be  infinite,  if  we  have  re- 
spectively 

1  K+i 
8Ci+8Ci«  — 4?— 4Ac=o,     or    c^= -— r, 

2  I  ~\~  ^^ 


CIRCULAR  CUBICS. 


205 


and 


8C2— 8c2/ci+4i— 4/c=o,     or    ^2  = 


1  K—t 

2  I—  /ci' 


so  that  the  equations  of  the  asymptotes  are 


(4) 


y  =  ix-{-  — 


I    K+i 


2   i-\- kV 
I    K—i 

y=  —  ix-\- :. 

•'  2    1—  Kl 


Solving  these  two  equations  simultaneously,  we  get  for  the 
coordinates  of  the  point  of  intersection  of  the  two  asymptotes  (4) 


(5) 


1  K^—  I 

2  K^+  I 


/C^+I 


The  sum  of  the  squares  of  these  expressions  is  x'^-]ry'^  =  \\  i.e., 
the  point  of  intersection  is  on  the  circle  corresponding  to  the 
infinite  line  in  the  Steinerian  transformation.  The  real  asymp- 
tote of  the  cubic  has  the  slope  k,  so  that  its  equation  is  of  the 
form  y=KX+C3.  By  a  similar  method  as  in  the  case  of  c^  and 
C2  above,  we  find 


c,= 


3«-« 


'^     1+2/C-' 
and  as  the  equation  of  the  real  asymptote 

3/c-/c^ 


(6) 


y=  KX-\- 


i+2k} 


Solving  (i)  and  (6)  simultaneously,  we  find  for  the  point  of 
intersection  of  this  asymptote  with  the  cubic 


(7) 


1  1—  K^ 

X^  ',      9 

2  I+ZC^ 


I+«2 


2o6  PROJECTIVE  GEOMETRY. 

Comparing  (7)  with  (5),  it  is  seen  that  the  two  points  are 
diametral. 

A  similar  result  is  obtained  by  taking  any  orthogonal  quad- 
ruple X  1^24  3^  4  and  the  circular  cubic  associated  with  it.  In 
this  case  the  equation  of  the  cubic  assumes  the  form 

(8)  {ax^^y){x'^-\-y'^)-^ax'^-{-2hxy-\-cy'^-\-2dx-\-2ey-\-j  =  o. 

Repeating  on  this  equation  the  same  process  as  above  on  equa- 
tion (i),  the  theorem  may  be  stated  thus: 

Considering  a  hicircular  cubic  in  a  Steinerian  transformation, 
the  asymptotes  at  the  circular  points  intersect  each  other  in  a  point 
D  0}  the  circle  which  corresponds  to  the  line  at  infinity  in  the  Stei- 
nerian transformation.  The  real  asymptote  cuts  the  same  circle 
in  a  point  C  which  with  D  determines  a  diameter  of  the  circle.  The 
points  D  and  C  are  called  center  and  principal  points  of  the  cubic. 

2.  In  equation  (8)  the  infinitely  distant  real  point  of  the  cubic 
is  the  infinite  point  of  the  line  ax-{-^y  =  o,  as  can  easily  be  verified. 
Taking  the  x-axis  parallel  to  this  line,  (8)  becomes 

(9)  y{x'^^y'^)-^ax'^-{-2bxy+cy'^-\-2dx-\-2ey-[-f  =  o. 

In  a  similar  manner  as  in  (5),  the  coordinates  of  the  center  of 
the  cubic  are  found  to  be 

a—c 

(10)  x=b,     >'  =  ^— . 

Taking  this  point  as  the  origin  of  a  new  coordinate  system 
with  axes  parallel  to  those  in  (9),  (9)  assumes  the  form 

(11)  {y-\-a){x'^-\-y'^)-{-2dx-{-2ey-\-j  =  o. 

Here  the  equation  of  the  real  asymptote  is  y=  —  a,  so  that 
the  coordinates  of  the  principal  point  of  the  cubic  become 

2ae—f 


CIRCULAR   CUBICS.  207 

Equation  (11)  may  be  considered  as  the  result  of  the  elimina- 
tion of  X^  from  the  two  equations 

x^-\-y^—  X^  =  o, 
2dx-]-2ey-\-j-\-X'^{y-\-a)=o, 

which  represent  two  projective  pencils  of  concentric  circles  and 
rays.     Hence  the  theorem  of  Czuber:^ 

Every  circular  cubic  may  he  generated  by  two  projective  pencils 
of  concentric  circles  and  rays.  The  common  center  of  all  circles 
of  the  pencil  is  the  center  of  the  cubic,  and  the  vertex  of  the  pencil 
of  rays  is  the  principal  point  of  the  cubic. 

3.  In  §  49  it  has  been  shown  that  the  points  on  a  cubic  may 
be  arranged  according  to  couples,  so  that  the  rays  from  any 
point  on  the  cubic  to  the  points  of  these  couples  form  an  involu- 
tion. 

Suppose  now  that  the  direction  of  the  real  asymptote  of  a 
circular  cubic  is  perpendicular  to  one  of  the  sides,  say  B2B3  of 
the  diagonal  triangle  B-^BJB^,  Fig.  86;  then  the  center  of  the 
cubic  will  coincide  with  the  point  B^.  In  other  words,  the  cir- 
cular points  form  a  couple,  so  that  the  involutions  of  rays  from 
any  point  of  the  cubic  contain  the  directions  of  the  circular  points 
as  a  pair. 

Hence,  according  to  a  theorem  in  §  5,  p.  21,  since  A^A^,  A^A^ 
are  two  couples  and  P  any  point  on  the  cubic,  the  angles  A^PA^ 
and  ^2-f^4  a-re  equal.     Hence  the  theorem: 

The  circular  cubic  which  contains  the  circular  points  as  a 
couple  {conjugate  pair)  is  the  locus  of  all  points  from  which  two 
■fixed  lines  A^A^,  A^A^  appear  under  the  same  angle. 

Inscribing  a  conic  to  the  quadrilateral  A^A2A3A^,  then  the 
pieces  A^A^  and  ^2^4  of  the  tangents  contained  between  the  two 
other  tangents  4 1^2)  ^3^4  of  the  conic,  are  subtended  by  equal 
angles  at  the  focus,  §  35,  p.   118.     Hence  the  theorem: 

'  Die  Kurven  drifter  und  vierter  Ordnung,  welche  durch  die  unendlich  fernen 
Kreispunkte  gehen.     (Zeitschr.  f.  Math.,  XXXII,  1887.) 


208 


PROJECTIVE   GEOMETRY. 


The  locus  of  the  foci  of  all  conies  inscribed  to  a  quadrilateral 
is  a  circular  cubic  having  the  circular  points  as  a  conjugate  pair. 

The  same  cubic  may  also  be  produced  by  two  projective  pencils 
of  circles  over  A^A^  and  A^A^,  in  which  two  corresponding  circles 
subtend  equal  peripheral  angles  over  the  chords  A^A^  and  A2Ai. 


Fig.  86. 


But  if  two  projective  pencils  of  circles  G+  XG^^o  and  G'+  XG^ 
=o  produce  a  cubic,  say  GG^—Gfi'=o,  so  that  this  equation 
reduces  to   Gg/—G-i^g'  =  o,   where   g/  and  g'   are  hnear  factors, 
then  the  cubic  may  also  be  produced  by  two  projective  pencils 
of  circles  and  rays. 

4.  In  the  same  circular  cubic  consider  the  pencil  of  circles 
through  B2BS,  Fig.  86.  The  ray  B^C  passes  through  the  center 
of  the  circle  through  B^B^B^.  A0A3  passes  through  the  center 
of  the  circle  through  ^2^3  (diameter  of  said  circle)  and  ^^-Bg. 
B^A^A^  passes  through  the  center  of  the  circle  through  A^A^^ 
(diameter)  and  B^Bs.      The  three  circles  through  ^2-^3  and  the 


CIRCULAR  CUBICS.  209 

three  corresponding  rays  through  B^  determine  nine  points  of  the 
cubic,  since  B^  as  a  point  of  tangency  on  BJJ  counts  twice.  The 
two  pencils  therefore  determine  two  projective  pencils  of  circles 
and  rays  whose  product  is  the  given  cubic.     Hence  the  theorem; 

The  circular  cubic  having,  the  circular  points  as  a  conjugate 
pair  is  also  the  product  of  a  pencil  0}  circles  and  a  projective  pencil 
of  rays  which  pass  through  the  centers  of  the  corresponding  circles. 

Ex.  I.  With  the  Steinerian  transformation  for  base,  prove  that 
the  general  equation  of  a  circular  cubic  has  the  form 

(ax-\-^y)(x^+y^)  +  ax^-\-2bxy+cy^+2dx-{-2ey+f  =  o. 

Ex.  2.  Given  the  pencil  of  circles 

x^+y^—  p^—2}.x  =  o     (^  =  constant) 

and  a  pencil  of  rays  passing  through  the  centers  of  these  circles. 
To  find  the  equation  of  the  product  of  the  two  pencils  and  discuss 
the  result  for  different  positions  of  the  vertex  of  the  pencil. 

Ex.  3.  Estabhsh  the  equations  of  two  projective  pencils  of 
circles  in  which  corresponding  circles  subtend  equal  peripheral 
angles  over  the  fixed  chords. 

Ex.  4.  Prove  that  in  a  circular  cubic  the  oval  and  the  ser- 
pentine appear  under  the  same  angle  from  any  point  of  the  curve. 

Ex.  5.  The  extremities  of  two  diameters  AiA^  and  A^A^ 
form  a  square.  What  is  the  locus  of  the  points  from  which  both 
diameters  appear  under  the  same  angle? 


§  51.   The   Five   Tjrpes  of  Cubics  in   the   Steinerian   Transfor- 
mation.^ 

I.  Cubic  with  Oval  and  Serpentine. 

This  cubic  is  obtained  when  all  four  points  of  the  funda- 
mental quadruple   are  either  real  or  imaginary.    As   the   case 

^  This  section  has   been  published  in  The  University  of  Colorado  Studies, 
Vol.  I.,  No.  4,  Feb.  1904. 


PROJECTIVE  GEOMETRY. 


with  four  real  points  has  so  far  always  been  used  to  illustrate  the 
general  properties,  we  shall  now  assume  an  entirely  imaginary 
quadruple  determined  by  a  coaxial  system  of  circles  with  the 
limiting  points  P  and  Q,  Fig.  87.     On  every  ray  g  through  an 


\^3 


Fig.  87. 


arbitrary  fixed  point  B  the  circles  of  this  system  cut  out  an  invo- 
lution of  points  whose  double-points  X  and  X'  are  two  points  of 
the  cubic  associated  with  the  point  B  in  the  Steinerian  transfor- 
mation belonging  to  the  given  imaginary  quadruple.  To  construct 
X  and  X',  let  g  cut  the  Une  w,  which  is  the  hne  joining  the  finite 
imaginary  points  of  the  quadruple,  at  M.  With  M  as  a  center 
pass  a  circle  K  through  P  and  Q  which  will  cut  g  in  the  required 


CIRCULAR   CUBICS.  211 

points.  In  reality,  according  to  the  well-known  construction 
just  explained  X  and  X'  are  the  points  of  tangency  of  g  with  two 
circles  of  the  given  system.  It  will  be  noticed  from  the  figure 
that  the  two  points  of  the  cubic  on  a  ray  through  B  are  always 
equally  distant  from  m.  Hence,  taking  a  ray  through  B  parallel 
to  m,  the  point  at  infinity  corresponding  to  Q  will  be  in  a  line 
through  P  parallel  to  m.  In  other  words,  the  line  through  P 
parallel  to  m  is  the  asymptote  of  the  cubic.  Considering  the 
pencil  of  circles  through  P  and  Q,  the  same  circular  cubic  is  also 
produced  by  this  pencil  and  the  pencil  of  diameters  through  B. 
Thus  a  statement  in  the  foregoing  section  is  corroborated. 

II.  The  Simple  Cubic. 

This  curve  is  produced  by  assuming  two  separate  real  and 
two  imaginary  points  in  the  fundamental  quadruple.  In  Fig.  87 
let  P  and  Q  be  the  real  points,  and  the  circular  points  of  the  pencil 
of  circles  through  P  and  Q  the  imaginary  points.  To  find  the 
points  Y  and  Y'  where  a  ray  I  through  B  cuts  the  cubic,  let  / 
cut  n  at  N.  With  iV  as  a  center  construct  the  circle  L  orthogonal 
to  the  pencil  of  circles  through  P  and  Q.  The  circle  L  cuts  I 
in  the  required  points  Y  and  Y' .  This  cubic  appears  again 
plainly  as  the  product  of  a  pencil  of  circles  and  the  pencil  of 
diameters  through  B.  Two  points  on  a  ray  through  B,  like 
Y  and  Y',  are  always  equally  distant  from  n.  To  R  corresponds 
the  infinitely  distant  point  of  the  cubic ;  consequently,  the  asymp- 
tote is  parallel  to  n  and  its  distance  SC  from  n  is  equal  to  RC. 

Ex.  I.  Prove  that  the  two  cubics  in  Fig.  87  intersect  each 
other  orthogonally. 

Ex.  2.  Construct  the  tangents  to  the  two  cubics  at  B,  P,  Q. 

III.  Cubic  with  an  Isolated  Point. 

The  quadruple  consists  of  two  distinct  points  A^,  ilg  and  two 
coincident  points  A2,  A^.  It  is  assumed  that  the  direction  of 
the  line  joining  A2  and  ^4  in  the  limit;  i.e.,  as  they  become  coinci- 
dent, cuts  A^As  at  B2.  Bi  and  B^  will  coincide  with  A  2  and  A^, 
Fig.  88.  Joining  B,  which,  as  usual,  has  been  assumed  infinitely 
distant,  to  B^,  B2,  B^,  and  constructing  the  fourth  harmonic  rays 
to  the  pairs  of  sides  passing  through  these  points,  it  is  seen  by 


212  PROJECTIVE  GEOMETRY. 

passing  over  to  the  limit  that  the  fourth  harmonic  rays  at  B^  and 
B^  coincide.  As  before,  they  cut  the  fourth  harmonic  ray  through 
B2  at  C,  the  point  through  which  the  asymptote  passes. 

The  pencil  of  conies  through  the  quadruple  cuts  every  ray 
through  B  to  the  left  of  ^3  and  to  the  right  of  ^1  in  elliptic  in- 


^■' 


/ 


/ 


1  ,'       -' 


volutions,  and  only  the  rays  between  A^  and  ^3  contain  hyperbolic 
involutions.  The  only  branch  of  the  cubic  is  therefore  contained 
between  two  lines  through  ^l^  and  A^  parallel  to  the  direction 
of  B.  The  ray  through  ^43^4  carries  a  parabolic  involution,  and 
A^A^  represents  an  isolated  point  of  the  cubic. 

IV.  Nodal  Cubic. 

Assuming  in  the  fundamental  qnadruple  Ai  and  A^  real  and 
coincident  and  A2,  A^  conjugate  imaginary,  a  cubic  with  a  double- 
point  or  node  at  A^A^  is  obtained.  In  Fig.  89  a  vertical  line 
through  A^A^  represents  the  limiting  direction  of  the  line  joining 
the  two  points.  As  conies  of  the  pencil  through  the  fundamental 
quadruple  take  the  pencil  of  circles  tangent  to  each  other  2X  A^A^ 
and  to  the  vertical  Hne.  yljand  A^  are  then  represented  by  the 
circular  points  at  infinity.  To  construct  the  cubic  associated  with 
an  arbitrary  point  B,  draw  rays  through  B.  On  each  of  these 
rays  the  pencil  of  circles  cuts  out  an  involution  whose  double- 


CIRCULAR   CUBICS. 


213 


points  are  points  of  the  cubic.  These  points  are,  of  course,  also 
the  points  of  tangency  of  circles  of  the  pencil.  Hence,  to  find 
the  points  where  a  ray  g  through  B  cuts  the  cubic,  take  the  point 
M  where  g  cuts  w,  the  Hne  joining  A^  with  ^4,  as  a  center  of  a 
circle  K  passing  through  ^1^4  4.  K  cuts  g  in  the  required  points 
X  and  X'.    From  this  it  is  seen  that  the  nodal  cubic  is  also 


the  product  of  a  pencil  of  circles  with  coincident  limiting  points 
and  a  pencil  of  diameters.  As  X  and  X'  are  equally  distant  from 
m,  the  asymptote  is  parallel  to  m  at  a  distance  to  the  left  of  m 
equal  to  BA^  {BA^l.  m  for  the  sake  of  symmetry.) 

V.  Cuspidal  Cubic.  > 

In  this  case  three  of  the  four  real  points  of  the  fundamental 
quadruple  coincide.  Constructively  this  can  be  arranged  best  by 
assuming  as  the  pencil  of  conies  a  pencil  through  a  fixed  point  A  ^ 
and  with  its  conies  all  osculating  each  other  at  another  fixed 
point  which  evidently  may  be  considered  as  the  representative 
of  the  three  coincident  points  A 2,  A3,  A^. 

To  construct  a  pencil  of  osculating  conies  we  may  start 
with  the  construction  of  §  41,  9,  Fig.  67.  There  it  was  shown 
that  the  picture  of  a  circle  in  a  perspective    collineation  whose 


214 


PROJECTIVE  GEOMETRY. 


center  lies  on  the  axis  of  collineation  and  also  on  the  given  circle 
is  a  conic  osculating  the  given  circle  at  the  center  of  collinea- 
tion. Hence,  considering  in  Fig.  90  the  line  5  joining  A^ 
with  the  coincident  remaining  points  as  the  common  axis  of  an 
infinite  number  of  perspective  coUineations    in  which  only  the 


Fig.  90. 


counter- axes   vary,   the   pictures    of    a    fixed    circle    K    through 
A-i^A^A^A^  form  clearly  a  pencil  of  osculating  conies. 

On  every  ray  g'  (or  the  identical  g/)  through  a  fixed  point  B 
(assumed  infinitely  distant)  this  pencil  cuts  out  an  involution 
whose  double-points  are  two  points  on  the  cuspidal  cubic  asso- 
ciated with  B  in  the  Steinerian  transformation.  These  points 
are  also  the  points  of  tangency  of  g'  (g/)  with  two  conies  of  the 
pencil.     For  the  actual  construction  of  these  points  the  following 


CIRCULAR  CUBICS.  215 

simple  method  may  be  applied:  Let  g'  intersect  s  at  S.  From 
S  draw  the  two  tangents  g  and  g^  to  the  circle  K}  Through  the 
center  of  collineation  or  the  cusp  draw  a  line  /  parallel  to  the 
direction  of  B.  Let  T  and  T^  be  the  points  of  intersection  of  I 
with  g  and  ^j,  and  through  T  and  T^  draw  two  lines  r  and  r^ 
parallel  to  s.  Considering  r  and  r^  as  counter-axes  of  two  dis- 
tinct coUineations  with  the  same  axis  5  and  the  same  center,  then, 
according  to  the  constructions  of  coUineations,  g'  and  g^  will  be 
the  pictures  of  g  and  g^  in  these  two  coUineations,  and  the  rays 
joining  ^  to  G  and  G^  cut  g'  (^/)  in  two  points  G'  and  G/  which 
evidently  are  the  points  of  tangency  with  g'  (g/)  of  the  two  oscu- 
lating conies  corresponding  to  the  fixed  circle  K  in  the  two  coUine- 
ations (r,  fi).  The  line  /  cuts  K  dX  U\  the  tangent  at  U  cuts  ^ 
at  F;  and  from  the  construction  follows  that  the  line  through  V 
parallel  to  I  is  an  asymptote.  In  a  similar  manner  the  lines  join- 
ing C  to  the  points  of  tangency  W  and  W^  of  the  tangents  to  K 
parallel  to  5  are  the  directions  of  the  two  other  real  asymptotes. 
By  a  suitable  collineation  this  cuspidal  cubic  may  be  transformed 
into  the  symmetrical  form  of  Newton's  parabola  cuspidata. 

Ex.  I.  Prove  that  if  5  is  a  diameter  of  K  and  the  direction 
of  B  is  perpendicular  to  s,  then  the  above  cubic  degenerates  into 
an   equilateral  hyperbola. 

Ex.  2.  Prove  that  if  K  is  tangent  to  s,  then  the  cubic  degener- 
ates into  a  parabola. 

Ex.  3.  A  pencil  of  cubics  is  determined  by  two  cubics  or  by 
eight  arbitrary  points  of  which  no  four  are  in  the  same  straight 
line.  But  it  is  clear  that  the  two  cubics  determining  the  pencil 
have  nine  points  in  common,  hence  all  cubics  of  the  pencil  pass 
through  these  nine  points.  In  other  words:  All  cubics  passing 
through  eight  -fixed  points  pass  through  a  ninth  fixed  point. 

Ex.  4.  Through  four  points  A,  B,  C,  Z>  of  a  cubic  draw  the 
pencil  of  conies  (iv).  Every  conic  K  of  this  pencil  cuts  the 
cubic  in  two  points  P  and  Q.  Prove  that  the  secant  PQ  cuts 
the  cubic  in  a  fixed  point. 

^  In  Fig.  90  K  is  the  only  circle  shown,  and  /  is  the  line  through  A  cutting 
this  circle  at  U. 


21 6  PROJECTIVE   GEOMETRY. 

Ex.  5.  Let  two  straight  lines  I  and  m  cut  a  cubic  in  the  points 
A,  B,  C  and  D,  E,  F.  Construct  the  points  of  intersection 
G,  H,  I  oi  AD,  BE,  CF  with  the  cubic  and  prove  that  they  are 
coUinear. 

Ex.  6.  Construct  the  cubic  in  the  Steinerian  transformation 
when  one  of  the  points  of  the  quadruple  is  infinitely  distant. 


CHAPTER  V. 

APPLICATIONS  IN  MECHANICS. 
§  52.   A  Problem  in  Graphic  Statics. 

I.  Let  I,  2,  3,  ...  be  a  system  of  coplanar  forces  in  a  plane, 
Fig.  91a.  With  O  and  O'  as  poles  construct  two  force-polygons, 
Fig.  gib,  and  in  the  previous  figure  the  two  corresponding  funicu- 
lar polygons.  Considering  in  both  figures  the  lines  o,  012,  0^12, 
o',  12,  it  is  seen  that  corresponding  fines  are  parallel  and  that 
they  form  in  each  case  five  sides  of  a  complete  quadrilateral. 
Hence,  according  to  the  last  theorem  of  §  25,  also  the  fine  joining 
the  intersections  of  o  and  o',  and  of  012  and  o'i2,  in  Fig.  91a,  is 
parallel  to  00'  in  Fig.  91^.  In  a  similar  way  it  can  be  proved 
that  the  line  joining  the  intersections  of  012  and  o'i2,  and  of  0123 
and  o'i23,  in  Fig.  91a,  is  parallel  to  00'  in  Fig.  gib,  i.e.,  identical 
with  the  line  joining  (o  and  o')  with  (012  and  0^12).  This  result 
may  evidently  be  extended  to ,  any  number  of  forces,  so  that  we 
have  the  theorem: 

Corresponding  sides  of  two  funicular  polygons  of  a  system 
of  coplanar  forces  intersect  each  other  in  points  of  the  same  straight 
line.  ^ 

Corollary. — //  the  forces  are  concurrent,  they  and  the  two 
funicular  polygons  determine  a  perspective  collineation. 

2.  The  value  of  this  theorem  will  appear  from  the  solution 
of  the  following  problem: 

Two  bars,  AC  and  BC,  connected  by  a  pivot- point  at  C,  are 
supported  by  pivots  at  A  and  B  (Fig.  92).      Two  forces,  i  and  2, 

*  In  Cremona's  Graphic  Statics  this  theorem  follows  from  the  fact  that  the  two 
figures  (a  and  h)  form  two  reciprocal  figures.     See  loc.  cit.,  p.  127. 

217 


2l8 


PROJECTIVE  GEOMETRY. 


are   applied  to  the  bar  AC,  and  in  the  same  manner  two  forces, 
J  and  4,  to  the  bar  BC.     To  find  the  reactions  at  the  points  A,B,C. 


Fig.  91a. 


Fig.  91&. 


First  determine  magnitude,  direction,  and  position  of  the 
resultants  (12)  and  (34)  of  the  forces  i,  2  and  3,  4  by  means  of 
the  polygon  of  forces  (Fig.  92b)  and  the  funicular  polygon 
(Fig.  92a).     Then  construct  the  funicular  polygon  of  the  result- 


APPLICATIONS  IN   MECHANICS. 


219 


ants  (12)  and  (34)  with  O'  as  a  pole  and  with  its  first  side  pass- 
ing through  A.  Every  funicular  polygon  constructed  in  this 
manner  will  be  collinear  with  every  other,  and  with  the  point 
of  intersection  M  of  (12)  and  (34)  as  the  center  of  perspective 


%/ 


/ 


'/\X 

A     ' 
/   / 


v// 


fR  = 


1234 


'';^><.— ___012         N      p/  \ 


Fig.  92a. 


Gollineation.  Now  it  is  clear  that  the  polygon  passing  through 
A ,  C,  and  B,  and  formed  by  the  reactions  at  these  points,  is  also  a 
funicular.  It  is  therefore  collinear  with  the  first  polygon  (o',  0^12, 
o'i234).  Projecting  the  points  C  and  B  from  M  upon  the  funicu- 
lar sides  (o'i2)  and  (o'i234),  respectively,  the  projected  points 
C  and  B'  will  correspond  to  C  and  5  in  a  perspective  collinea- 
tion.     Hence  the  Hnes  BC  and  B'C  will  intersect  each  other 


220 


PROJECTIVE  GEOMETRY. 


in  a  point  S  of  the  perspective  axis  s.      By  this  point   and  the 
point  A  the  axis  is  determined. 


The  directions  of  the  reactions  at  ^,  C,  and  B  intersect  the 
funicular  hnes  o',  o'i2,  0^1234  in  points  of  the  Hne  s,  and  they 
"may  easily  be  drawn.  To  find  the  magnitudes  of  the  reactions, 
draw  lines  parallel  to  their  directions  through  the  points  A^,  C^, 
Bi-  These  lines  necessarily  meet  in  a  point  O^  of  the  straight 
line  g'.  Thus  O^A^,  O^B^,  O^C^  are  the  magnitudes  of  the  reac- 
tions at  the  points  A,  B^C. 


§  53.   Statical   Proofs    of   Some    Projective    Theorems. 

I.  Constructing  a  funicular  polygon  of  a  system  of  coplanar 
forces  I,  2,  3,  .  .  .  ,  w,  it  is  known  that  the  resultant  of  the  system 
passes  through  the  point  of  intersection  of  the  forces  (o)  and 
(0123  . .  .n)  of  the  funicular  polygon,  and  is  also  the  resultant 


APPLICATIONS  IN  MECHANICS.   ,  2  21 

of  these  two  extreme  forces,  with  (o)  reversed.  Hence,  when 
the  system  is  in  equihbrium,  the  forces  (o)  and  (0123  .  .  .  n) 
must  coincide.     Hence  the  theorem: 

A  funicular  polygon  of  a  system  of  coplanar  forces  in  equilibrium 
is  a  closed  figure. 

Consider  now  three  forces  i,  2,  3  in  equihbrium.  Fig.  93a,  and 
draw  any  triangle  o,  01,  012  having  its  vertices  on  these  forces. 
Draw  also,  in  Fig.  936,  the  force-polygon  123.     Through  the  inter- 


FiG.  93&.  Fig.  93a. 

section  of  i  and  3  in  (b)  draw  a  line  parallel  to  o  in  (a) ;   through 

1  and  2  one  parallel  to  ci  in  (a).  These  two  lines  intersect  each 
other  in  a  point  O.      Now  connect  O  wdth  the  intersection  of 

2  and  3.  Thus  three  forces  OA,  OB,  OC  are  obtained,  and 
if  O  is  assumed  as  a  pole  and  starting  out  with  the  original  hne 
o  in  (b),  a  funicular  polygon  is  obtained  which  coincides  with  the 
original  triangle  o,  01,  012.     Hence  the  theorem: 

Any  triangle  whose  vertices  lie  on  the  lines  of  action  of  three 
forces  in  equilibrium  may  be  considered  as  a  funicular  polygon 
of  these  forces. 

Consequently,  according  to  the  theorem  of  §  51,  if  we  take 
any  two  triangles  with  their  vertices  on  the  three  lines  of  action, 


2  22  PROJECTIVE   GEOMETRY. 

the  three  points  of  intersection  of  corresponding  sides  are  colHnear. 
As  any  three  concurrent  Hnes  may  be  chosen  as  Hnes  of  action 
of  three  forces  in  equihbrium,  we  thus  have  proved  the  well- 
knowTi  theorem  concerning  homologous  triangles. 

2.  Theorem.     The  middle  points  of  the  diagonals  of  any  com- 
plete quadrilateral  are  collinear.^ 

IMiNCHiN  ^  gives  the  following  proof  for  this  proposition :  Let 
ABCDEF  be  the  complete  quadrilateral.     Take  the  following 

system  of  forces,  supposed  act- 
ing on  a  rigid  body:  two  forces 
represented  by  DA  and  DC  in 
magnitudes  and  senses,  and 
two  represented  by  BA  and  EC, 
Fig.  94 

Now    the    resultant    of    the 

first  pair  passes  through  a,  the 

middle  point  of  .4C;    so  does 

Fi^-  94-  the    resultant    of    the    second 

pair;  therefore  the  resultant  of  the  four  forces  passes  through  a. 

Also  the  resultant  of  DA  and  BA  passes  through  ^,  the  middle 

point  of  BD;    so  does  the  resultant  of  DC  and  BC;    hence  the 

resultant  of  the  four  forces  also  passes  through  /?.     We  shall  now 

show  that  it  passes  through   y,  the  middle  point  of  EF.     For 

this  purpose  introduce  a  force  ED  and  a  force  DE  which  do  not 

alter  the  given  system.     Introduce  also  forces  CE,  EC ;  CF,  FC ; 

FB,  BF.     Hence  the  given  system  is  equivalent  to  forces  EA, 

AE;   DF,  DE;    BB,  BE;   EC,  FC;    and  it  is  obvious  that  the 

resultant  of  each  of  these  pairs  passes  through   y;    hence  the 

resultant  of  the  whole  system  passes  through   y.     Now  as  the 

resultant  of  the  given  system  acts  in  a  right  line,  and  as  a,  /?,  y 

have  been  independently  shown  to  be  points  on  this  resultant, 

these  points  are  collinear.     q.e.d. 

3.  Pascal's   theorem,   that   the   intersections   of  the   opposite 


^  Chasles,  loc.  cit.,  arts.  344,  345. 

*  Treatise  on  Statics,  Vol.  I,  pp.  145,  146. 


APPLICATIONS  IN  MECHANICS. 


223 


AF 
DC 

two 


sides  of  a  hexagon  inscribed  in  a  circle  lie  in  a  right  line,  is  easily- 
exhibited  as  a  case  of  the 
funicular  property  in  §  52. 
Following  again  Minchin,  loc. 
cit.,  let  the  lines  DA,  EB,  FC 
in  Fig.  95  be  lines  of  action 
of  three  forces,  P,  Q,  R,  such 
that  if  P  is  resolved  at  A  into 
two    components    along   AB, 

or  into  two  at  D  along 

DE ;  if  <2  is  resolved  into 

at  B  along  BA,  BC,  or 
into  two  at  E  along  ED,  EF; 
and  if  R  is  resolved  at  C 
along  CB,  CD,  or  at  F  along  /-' 
FE,  FA,  the  two  compo- 
nents    thus     obtained     along 

any  side  are  equal  and  opposite.  Obviously  such  conditions  are 
consistent,  on  account  of  the  equality  of  angles  in  the  same  seg- 
ment of  a  circle.  Now  if  P,  Q,  R  are  applied  at  ^,  B,  C,  by 
the  nature  of  the  case  a  polygon  FA  BCD  of  jointed  bars  pivoted 
at  F  and  D  would  be  kept  in  equilibrium;  i.e.,  this  is  a  funicu- 
lar of  the  forces.  Again,  let  P,  Q,  R  be  applied  at  D,  E,  F  to 
a  polygon  CDEFA  of  jointed  bars  pivoted  at  C  and  A.  This 
polygon  would  be  in  equilibrium,  and  a  funicular  of  the  forces. 
The  two  funiculars  of  the  same  forces,  however,  have  the  property 
that  the  intersections,  a,  /?, ;-,  of  their  corresponding  sides  (AB,  DE), 
(BC,  EF),  {CD,  FA)  are  collinear.     q.e.d. 

Ex.  Prove  that  the  medians  of  a  triangle  are  concurrent. 


Fig.  95. 


Geometry  of  Stresses  in  a  Plane. 


§  54.   General  Remarks. 

Forces  acting  on  a  body  cause  certain  displacements  or  strains 
between  its  particles.    These   strains  are  said  to  be  within  the 


224  PROJECTIVE   GEOMETRY. 

elastic  limit  if  after  the  disappearance  of  these  forces  the  strains 
disappear  also;  i.e.,  if  the  body  returns  to  its  original  condition. 
The  forces  which  occur  within  the  body  as  a  result  of  the  strains 
are  called  stresses.  These  are  called  tensions,  compressions,  or 
shears,  according  as  their  tendency  is  to  pull  the  particles  apart, 
to  press  them  together,  or  to  push  them  over  one  another.  Accord- 
ing to  Hooke's  law  the  stresses  in  a  body  are  approximately  pro- 
portional to  the  corresponding  strains  as  long  as  they  occur  within 
the  elastic  limit.  In  many  cases,  plane  surfaces  may  be  passed 
through  strained  bodies  orthogonally  to  which  there  are  no 
strains  and  consequently  no  stresses.  This  is  the  case  in  beams 
under  tension,  compression,  or  bending,  and  covers  a  great  num- 
ber of  engineering  structures.  In  these  cases  the  investigation 
of  strains  and  stresses  is  limited  to  the  plane.  In  what  follows 
only  stresses  in  a  plane  will  be  considered. 

The  forces  producing  the  stresses  in  a  body  and  these  them- 
selves are  in  equilibrium.  The  stresses  in  any  portion  of  the 
solid  are  also  in  equilibrium.  Considering  an  infinitesimal 
plane  section  in  a  strained  solid,  we  make  the  assumption  that 
the  stresses  acting  on  this  element  are  uniformly  distributed, 
so  that  their  resultant  passes  through  the  center  of  gravity  of 
this  element.  For  many  purposes  it  is  convenient  to  consider 
the  resultant  stress  per  unit  of  the  surface-element.  This  stress, 
the  resultant  divided  by  the  element,  is  called  the  specific  stress 
acting  on  that  element. 

§  55.   Involution   of  Conjugate   Sections  and  Stresses. 

I.  Calling  a  plane  surface  through  a  strained  body  with  the 
stresses  acting  in  this  pla,ne  (no  stresses  normal  to  the  plane,  as 
assumed  above)  a  field  of  forces,  we  assume  that  under  the  influ- 
ence of  this  field  every  portion  of  this  plane  is  in  equilibrium. 
Thus,  if  a  very  small  triangle  ABC  (infinitesimal  in  all  rigor) 
is  cut  out  of  the  field,  the  resultant  stresses  acting  upon  its  sides 
must  be  in  equilibrium.  According  to  the  assumption  of  the 
uniform   distribution   of   stresses   over   an   infinitesimal   section, 


APPLICATIONS  IN  MECHANICS. 


225 


these  resultants  must  pass  through  the  middle  points  y,  a,  ^  of 
the  sides  AB  =  c,  BC  =  a,  CA=b,  and,  being  in  equilibrium,  are 
concurrent.  Designate  these  resultants  by  ^,  B,  C,  as  shown 
in  Fig.  96.  Each  two  of  these  forces,  for  instance  A  and  B,  may 
be  resolved  into  components  parallel  to  the  sides  AC  and  BC. 
Let  A^,  A2  and  5^,  B2  be  these  components.  A  2  and  B^  act  along 
the  sides  BC  and  AC,  while  A,\\AC  and  B^  \\  BC.  As  Aj^  and  B^ 
both  pass  through  ;-,  their  resultant  will  pass  through  y. 


Fig.  96. 

In  consequence,  the  resultant  of  ^2  3-nd  B2,  which  passes 
through  C,  must  pass  through  y,  since  C  is  the  resultant  of  ^j,  B^ 
and  A  2,  B2.  Now  C;-  is  half  the  diagonal  in  the  parallelogram 
having  AB  as  the  other  diagonal.  In  order  that  the  resultant 
of  ^2  s-iid  ^2  lies  in  the  diagonal  Cf  the  proportion 


■  b 


must  hold.     —  and 
a 


B2 

-r,  however,  are  the  specific  stresses  acting 

along  the  sections  BC  and  AC.     Hence  the  theorem: 

//  the  specific  stresses  acting  on  two  different  sections  at  a  point 
are  each  resolved  into  components  parallel  to  these  sections,  then 
the  components  acting  along  these  sections  are  equal. 


226 


PROJECTIVE  GEOMETRY. 


If  the  sections  AC  and  BC  are  perpendicular  to  each  other, 
A-^  and  B^,  A^  and  B^  represent  the  normal  and  transversal  com- 
ponents of  the  stresses.     Hence  the  corollary: 

The  transversal  stresses  acting  on  two  perpendicular  sections 
are  equal. 

Taking  at  C  any  section  CB  and  the  stress  A  acting  on  it  and 
drawing  another  section  CA  parallel  to  ^,  we  have  A^=A,B^  =  B, 
4  2  =  ^2  =  o-     This  gives  the  corollary : 

//  the  force  A  acting  on  a  section  CB  is  known,  then  the  force  B, 
acting  on  a  section  CA  parallel  to  A ,  is  parallel  to  CB. 


Fig.  97. 

2.  According  to  this  corollary  for  every  section  through  a 
point  C  there  exists  a  certain  stress  acting  on  this  section,  such 
that  if  the  direction  of  this  stress  is  considered  as  a  section,  the 
stress  belonging  to  this  section  acts  along  the  original  section. 
This,  however,  is  a  clear  expression  for  the  involutoric  character 
of  the  directions  of  sections  and  corresponding  stresses. 

To  prove  this  directly,  keep  in  Fig.  97  the  stress  A  and  the 


APPLICATIONS   IN  MECHANICS.  227 

direction  of  the  stress  B  ||  CB  constant,  and  let  the  section  BA 
turn  about  the  fixed  point  B.  The  extremity  A  of  BA  traces 
on  the  section  CA  a  point-range  AA^A^ .  .  .  ,  so  that  the  corre- 
sponding stresses  B,  B^,  B^.  .  .  are  proportional  to  the  distances 
CA,  CAj^,  CA2,  ....  In  the  force-polygon  the  extremities  of 
the  -B-stresses  are  marked  by  LL^L 2 .  .  .  ,  and  the  corresponding 
C-stresses  are  ML,  ML^,  ML^,  ....  Now  the  distances  NL,  NL^, 
NL2,  .  .  .  are  proportional  to  the  distances  CA,  CA^,  CA^,  •  •  .  ; 
hence  the  projectivity  of  the  pencils 

{B-AA^A2...)7^{M-LL^L2...). 

Moving  these  pencils  parallel  to  themselves  so  that  M  coin- 
cides with  B,  we  have  at  B  an  involutoric  pencil  of  sections  and 
directions  of  corresponding  stresses. 


§  56.    Discussion  of  this  Involution. 

In  Fig.  97  the  sections  CA  and  CB  are  both  acted  upon  by 
compressions;  in  consequence  the  stress  acting  on  the  section  is 
a  compression.  From  the  figure  it  appears  clearly  that  corre- 
sponding rays  of  the  involution  in  this  figure  move  in  the  same 
direction.  Hence,  according  to  §  3,  the  involution  is  elliptic. 
The  same  is  true  if  there  are  only  tensions.  In  these  cases  there 
are  no  double-rays,  i.e.,  there  are  no  sections  where  there  are  only 
shearing  {transversal)  stresses.  In  all  sections  the  material  is 
either  entirely  under  the  influence  of  compressions  or  Under  the 
influence  of  tensions. 

As  every  involution  in  a  pencil  admits  of  two  rectangular  rays, 
it  follows  that  through  every  point  of  a  plane  of  stresses  there  are 
two  sections  on  which  only  normal  stresses  are  acting.  In  case  of 
elliptic  involutions  these  normal  stresses  are  either  both  compres- 
sions or  both  tensions. 

If  in  Fig.  97  one  section,  say  CA,  is  acted  upon  by  a  tension 
and  the  other,  CB,  by  a  compression,  it  is  apparent  that  corre- 
sponding rays  of  the  involution  (§  3)  move  in  opposite  directions. 


228 


PROJECTIVE   GEOMETRY. 


The  involution  is  hyperbolic  and  has  two  real  double-rays 
(sections)  in  which  only  shearing  stresses  are  acting. 

Considering  two  corresponding  rays  BA  and  C,  they  are  al- 
ways separated  by  one  of  the  double-rays,  say  d^,  Figs.  98  and  99. 
If  a  compression  acts  on  BA ,  it  will  be  so  when  BA  approaches 


Fig.  99. 


d^.  But  as  soon  as  these  corresponding  rays  have  crossed  the 
double-ray  d^,  the  section  AB  is  acted  upon  by  tension.  From 
this  it  follows  that  the  material  included  by  one  angle  formed  by 
the  double-rays  is  subject  to  tension  only,  while  the  supplementary 
part  is  subject  to  compression  only. 

As  the  angles  formed  by  the  double-rays  are  bisected  by 
the  rectangular  pair,  it  follows  that  the  stress  acting  on  one  section 
where  there  is  only  a  normal  stress  is  a  compressive  force,  while 
the  stress  acting  on  the  perpendicular  section  is  a  tensile  force. 

For  a  further  discussion  of  these  involutions  and  their  exten- 
sion to  space  we  refer  to  Ritter's  Graphische  Statik,  Vol.  I,  pp. 
1-46,  pubhshed  by  Meyer   &  Zeller,  Zurich. 


APPLICATIONS  IN  MECHANICS. 


229 


I  57.  The    Stress    Ellipse.^     Metric    Properties    of   the 
Involution  of  Stresses. 

I.  According  to  the  previous  section  the  specific  stress  acting 
on  every  section  through  a  fixed  point  can  be  constructed  as  soon 
as  the  specific  stresses  acting  on  any  two  sections  are  known. 
In  Fig.  100  assume  these  two  sections,  CA  and  CB,  parallel  to  the 
X-  and  ;y-axis  of  a  Cartesian  system,  and  let  the  variable  section 
AB  include  an  angle  a  with  the  positive  part  of  the  x-axis.  As 
in  Fig.  96,  resolve  the  stresses  A   and  B  into  transversal  and 


Fig.  100. 

normal  components  A^,  Bo  and  A^,  B^.  Designating  the  specific 
stresses  determined  by  these  components  by  t^,  /^  and  n^,  \,  we 
have  Aj^  =  a-7t^,  Bi  =  b-ni^,  A2  =  a-t^,  B^  =  b-ti^.  Evidently  ^„  =  ^^  = /, 
say.  The  resultant  C  of  ^  and  B  can  also  be  resolved  into  trans- 
versal and  normal  components  T  and  A^  with  the  corresponding 
specific  stresses  t  and  v,  so  that  T  =  cr,  N  =  cv.  Let  X  and  Y 
be  the  components  of  C  parallel  to  the  coordinate  axes.     Desig- 


^  Elegant  graphic  constructions  for  stress-ellipses  may  be  found  in  Ritter's 
Graphische  Statik,  loc.  cit.  For  a  thorough  analytic  discussion  see  M.  Levy's 
Statique  Graphique,  Vol.  I,  pp.  527-548  (Note  IV). 


230  PROJECTIVE  GEOMETRY. 

nating  the  specific  stresses  by  ^  and  f],  X  =  c^,  Y=ct).     From  the 
figure  we  have 

a  =  c  sin  a,     b  =  —  c  cos  a, 
X=A^+B2  =  an^-\-b-i^,     or 
X  =  c{n^  sin  a—t^  cos  a)     and 

^  =  n^  sin  a—tf^  cos  a. 
Y--B^^-A^  =  hn^-Vat^,     or 
Y  =  —  c{n^  cos  a  —  ^o  sin  a)     and 

Tj  =  —  Wj  cos  a  +  ^„  sin  a. 

Now 

v  =  — 7;  cos  a+6  sin  a,     hence 

v=n,  cos^  a—t   sin  a  cos  a  +  w„  sin^ «— ^h  sin  a  cos  a,     or 

^  =  K^a+ ^J  +  i  cos  2a  •  (w^- w  J  -  i  sin  20:  •  (/^+ /  J. 

Similarly, 

T  =  (f  cos  a  +  rj  sin  a,     or 

z  =  n^  sin  a  cos  a— ^^  cos^  a  — w^,  sina  cos  a  +  /^  sin^  a,     or 

r  =  i(M,-w,)  sin  2a+K^a-0-K^a+0  COS  2a. 

As  t^  =  t^  =  t,  we  have  finally 

(i)  y  =  KWa+^6)-i(^a-%)  cos  2a-/  sin  2a, 

(2)  'r  =  K^o~"^fe)  ^^^  20:—/  cos  2a. 

Designating  the  angle  which  the  direction  of  C  makes  with 
the  positive  x-axis  by  /?,  we  have 

Tj      —fif^cos  a  +  t^  sin  a 
^^P~ ^  ~  n^sina—ti^  cos  a  ' 

or 

/  tan  a—n. 


tan/?  = 


n„  tan  a  —  t 


APPLICATIONS  IN  MECHANICS.  231 

Solving  for  tan  a,  we  get 

(3)  tan  a 


/  tan  /?—  n^ 


n^  tan  /?—  t ' 

which  clearly  shows  the  involutoric  character  between  the  direc- 
tions of  a  section  and  the  stress  acting  on  this  section.     This  is 
in  agreement  with  the  geometric  discussion  of  stresses  in  §  55. 
For  the  double-elements  of  this  involution  we  have 

tan  |5  =  tan  a=m, 

(4)  m= -— ^-'. 

These  values  for  m  are  real  when  n^  and  n^  have  different 

signs;    in  this  case  the  involution  is  hyperbolic.     If  n^  and  n^ 

both  have  the  same  sign,  and  n^n^yf^,  then  the  involution  is 

elliptic.      For  n^^  =  f  the  involution  is  parabolic  and  tan  a  = 

t  . 

—  =  const.;  i.e.,  the  stresses  all  act  in  the  same  constant  direction. 

a 

This  is  the  case  in  a  rod  under  tension  or  compression  exclu- 
sively. 

2.  Letting  v^  and  v^  b^  the  normal  specific   stresses  on  two 

perpendicular  sections  determined  by  the  angles  a  and  a , 

and  it  T„  the  transversal  specific  stresses  in  these  sections,  from 
the  formulas  for  v  and  r  we  get 

T^^— ^1^2  = /^— w^w^  =  const. 

To  get  the  rectangular  pair  of  the  involution,  we  form 

^tana  — w. 


tan  a  — 


/        ns  w^tana— ^ 

tan  (a-i9  =- f-^ •. 

/tana— w, 

i  +  tan  a — : 

w„tana— f 


232  PROJECTIVE   GEOMETRY. 

^71  /  tan  oi—n. 

In  this  a—p=—,  if  i  +  tana : ,  =  0,   or 

'2  n^  tana— ^ 

/  tan^a+(w^— wj  tan  a— /  =  o,     or 

tan  a  =  -^ ^^ — ^^ °^      ^  , 

an  expression  which  is  always  real. 

From  this,  and  also  from  the  expression  for  7  =  0,  follows 

2t 

(k)  tan2Q:= . 

3.  We  shall  now  find  the  locus  of  the  extremities  of  the  specific 

C        . 

stresses  —  acting  on  all  sections.     Its  coordinates  are  evidentl_v 

c  and  T)  when  referred  to  the  point  of  appHcation  as  an  origin. 
From 

(6)  n^  sin  a  —  ^^  cos  a  ==  6, 

(7)  t^  &m  a  — fif^  cos  a  =  7j, 
the  expressions  for  sin  a  and  cos  a  result : 


sm  a  = 


n^n^-f' 


cos  a  = To 

a    b 

and  since  sin^  a  +  cos^  «  =  i,  the  required  equation 

(8)         P(V+0-2erjK+w,)/+7;^(V+^')-(V6-0'=o 

results.     In  this  equation 

it  represents,  therefore,  an  ellipse,  the  so-called  stress-ellipse. 


APPLICATIONS  IN  MECHANICS.  233 

From  analytical  geometry,  §  31,  the  angles  d  and  /9+— of  the 

axes  of  this  elhpse  are  determined  by 

—  2t(n^-\-n^)             2t 
(9)  tan  2(9= — 5-—^ — 7 — r7~m  = • 

Hence,  according  to  (5),  we  have  the  theorem: 
The  axes  of  the  stress-ellipse  coincide  with  the  rectangular 
pair  0}  the  involution  of  stresses  around  the  center  of  the  ellipse. 
From 

^""K^a+^6)~K^a~^&)  ^0^  20:— /sin  2a 

we  find,  by  differentiation  with  respect  to  a,  the   condition  for 
maximal  and  minimal  normal  stresses, 

/  sin  2o:(Wj— w^)—  2/  cos  20:  =0, 
or 

2t 

(16)  tan  20:  = . 

u^—n. 

a  0 

Hence,  according  to  (5),  the  theorem: 

The  maximal  and  minimal  normal  stresses  occur  on  the  sections 
of  the  rectangular  pair  of  the  involution,  or  on  the  axes  of  the  stress- 
ellipse.     In  these  sections  r  =  o. 

In  a  similar  manner,  from 

T  =  J(w^— Wj,)  sin  2a— t  cos  2a 

we  find  for  the  maximal  and  minimal  shearing  stresses  the  condi- 
tion 

n  —n, 
(11)  tan2a'  =  — — ; 

hence,  from  (10)  and  (11), 

Tt  71 

tan  2Q:-tan  2a  =  — i,     2a  =  2a,:^—,     «=«£—,     or: 

2  4 

The  directions  of  the  maximal  and  minimal  shearing  stresses 
bisect  the  angles  formed  by  the  maximal  and  minimal  normal 
stresses  and  are  equal  (except  as  to  sign). 


234  PROJECTIVE   GEOMETRY. 

4.  The  directions  /?i  and  /?2  of  the  stresses  corresponding  to 

two  rectangular  sections  with  the  incHnations  a  and  a-\-—  are, 

2 

according  to  (3),  determined  by 

/     X  ^      i  tan  a  —  n, 

(13)  tan^2=! 


t  cot  a  +  % 
w„cot  a+^' 


^  ^     ^  ■    /tan/?i— w,  w^tan/?,  — / 

From  (12),     tan  q:  =  — : ^ — ,,     cota  =  .- ^ . 

^  w^  tan  p^—  r  ^  tan  p^—  n^ 

Substituting  this  in  (13)  and  reducing,  we  get 
(14)  ^^^^^      /K+-.)tanA-(V+/^) 


(w/+^2)tan^,  -^K+wJ' 

From  this  formula  follows  at  once : 

The  directions  of  pairs  0}  stresses  corresponding  to  pairs  of 
perpendicular  sections  form  an  involution. 

This  involution  is  identical  with  the  involution  of  conjugate 
diameters  of  the  stress-ellipse. 

To  prove  the  second  part  of  this  theorem,  form  the  equa- 
tion of  the  polar 

(l5)^U<+n-i^^V+^V^)(n^+n,)i+VV^K+n-Kn,-t'r=o 

for  any  point  (l"i,  r^J,  for  which  -^  =  tan/?i,  with  respect  to  the 

stress-ellipse. 

For  the  point  infinitely  distant  in  the -direction  of   /?i  there 

still  is  -^  =  tan  /3i  and  1^1=  00,  7ji=  00.     Hence 
''1 

H<+n-  (V+^  tan  A)K+ w,)  +  )j  tan  l^,{n^'+t')  =0, 


APPLICATIONS  IN  MECHANICS.  235 

or 

Ti       /(,,^+;zJtan/?,-(V+0 


which  is  nothing  else  than  tan  /?2  in  (14),  q.e.d. 

5.  The  normal  stresses,  Vj,  Vj,  on  two  perpendicular  sections, 

determined  by  the  angles  a  and  a ,  are 

i^i  =  i(^^a+'0~  2(^a"~^0  cos  2a— t  sin  20;, 
^2=2('^a+"fc)  +  K^^a~"b)  ^os  20:  +  ^  sln  20:. 

Adding,  we  get 

(i6)  !^i+v2  =  w^+«^  =  const. 

Hence  the  theorem: 

The  sum  of  the  normal  stresses  acting  on  two  perpendicular 
sections  is  constant  and  equal  to  the  sum  of  the  maximal  and  mini- 
mal normal  stresses. 

6.  Between  the  angles  ,5  and  a  which  the  directions  of  a  sec- 
tion and  the  corresponding  stress  make  with  the  positive  ic-axis 
the  involutoric  relation 

.     .  r,     I  tan  a—n^ 

(17)  tan  .5  =  ; 


n^  tan  a  —  t 


exists.  The  central  ray  of  the  involution,  for  which  ^=0,  is  deter- 
mined by  tan  a  =  y.     Designate  this  value  of  a  by  y,  so  that 

n, 

—  =  tan  y.     Take  a  line  parallel  to  the  ray  for  which  /?  =  o,  at  a 

distance  p  from  it,  and  project  the  involution  of  stresses  on  this 
line.  Then,  from  Fig.  loi,  ^C-5C  =  const.  To  determine  this, 
constant,  we  have  BC  =  BD—CD,  AC  =  CD-\-DA\   hence 


AC-BC  = 


/__i p  \(  p ^\ 

\  tan  ^     tan  yj  \  tan  y     tan  a/ ' 


236  PROJECTIVE  GEOMETRY. 

or,  after  reducing, 


<i8) 


n  n,  —  P 
AC-BC  =  -p^  "  \     =k,  say. 


In  a  similar  manner,  for  the  constant  of  the  involution  of 
conjugate  diameters  of  the  stress-ellipse  we  find 


(19) 


(fl    ft    —  t     I 

A,C,-B,C,  =  -p^^-^^p-:^=K,  say. 


Assuming  that  w„  and  n^  are  normal  stresses  on  two  perpen- 
dicular sections,  then  t  =  o.     Without  loss  of  generality  we  may 


p  =  o 

y 

1 

\ 

P 

\ 

X^ 

4 

I 

V^ 

C     D 
Fig.  ioi. 


also  assume  p  =  T-,  so  that  (18)  and  (19)  become 


where  n^  and  w^  now  designate  the  maximal  and  minimal  normal 
stresses.  Hence,  when  the  stress-ellipse  is  known,  it  is  not  diffi- 
cult to  construct  the  involution  of  stresses. 


§  58.  Examples. 
I.  A  plane  linear  deformation  is  defined  by  the  equations 


<i) 


:</  =  ax-\-hy, 
y'  =  a^x+h^y. 


APPLICATIONS  IN  MECHANICS.  237 

Referring  the  points  {x,  y)  and  {x' ,  y')  to  an  oblique  system 
of  coordinates  (I",  rj)  having  the  same  origin  and  whose  axes 
include  the  angles  a  and  a^  respectively,  we  have 

( :v  =  6  cos  a-\-rj  cos  a^, 
(2)  \ 

(  3'  =  c  sm  a+ 5y  sm  «!• 

Applying  this  to  the  points  (.v,  y)  and  {x' ,  /),  we  get,  accord- 
ing to  (i), 

^'  cos  a+  f)'  cos  a^  =  ^{a  cos  a+h  sin  «)  +  7j(a  cos  «!+&  sin  a^), 

f  sina+7}'  sin  a'i  =  6(ai  cos  a-\-hiSm.a)  +  rj{a^  cos  a^+^i  sinaj, 

or,  solving  for  f'  and  j;', 


^'  =  -; — 7 ;  I  f  (d  COS  a  sin  a,  +  6  sin  a  •  sin  a,  —  a,  cos  o;  cos  a^ 

sm  («!— a)     ^  ^      ^  ^ 

—  &i  sin  a  cos  a^  +  7;[6  sin^  «!—  a^  cos^  a^ 
+  (a—  6  J  sin  a^  cos  a^]  \ , 

77'=  ^ — -, \\^\h  sin^ai— fli  cos^a;+  (a—h,)  sin  a  cosal 

'      sm  (a— tti)'  V        1/  J 

+  jj(fl  cos  «!  sin  a  +  i  sin  a^  sin  a—a^  cos  a^  cos  a: 

—  &i  sin  CKi  cos  a)  | . 

In  these  expressions  we  can  choose  the  angles  a  and  a^  in 
such  a  manner  that  tan  a  and  tan  a^  are  the  roots  of 


so  that 


&-tan^  a+ (a— &J  tana— ai  =  o, 
h^—  a-\-\/  {h^—  ay -\- ^aj) 


tan  a  = 


26 


2^8  PROJECTIVE  GEOMETRY. 

Under  these  conditions  the  coefficients  of  jj  and  <?  in  the  ex- 
pressions for  ^'  and  t]' ,  respectively,  vanish  and  the  hnear  trans- 
formation in  this  system  of  obhque  coordinates  assumes  the  form 

C  ^,_  - 2{a\-- 2a,h)^-{a'+h^^)+{a-\-\W{h^-ay+^aJ 


L  V  (61 -a)  =^  +  4^1^ 

From  these  formulas  follows  that  the  linear  deformation  (i) 
may  be  considered  as  two  consecutive  stretches  along  two  oblique 
axes  or  directions.  The  angle  (jS  formed  by  these  axes  is  deter- 
mined by 

2.  Evidendy  the  rectangular  transformation 

y'-Ky 

is  a  special  case  of  (3). 

The  elongations  along  the  x-  and  >'-axes  are 

x'—x  ,    y'~y    7 

=  a— I     and     =  0,-1. 

X  y 

We  can  also  write  (4)  in  the  form 

r  x'  =x+  {a—  T-)x, 
■■(5)  [y=y+ib-i)y. 

Consider  (5)  as  the  analytical  expression  of  a  strain  in  a  very 
thin  plate  which  is  -assumed  to  have  the  propert}'  of  a  perfect 


APPLICATIONS  IN  MECHANICS.  239 

solid.      Then  a—  i   and  ^j—  i   are  very  small  numbers.      The 
strain-ellipse  has  the  equation 

By  the  linear  deformation  certain  stresses  are  produced  which 
according  to  Clebsch  ^  may  be  expressed  in  terms  of  the  strains 
a—  I  and  h^—i.  In  our  special  case  there  are  no  shearing  stresses 
along  the  x-  and  }'-axes,  so  that  in  the  formulas  ^ 

we  have  (9  =  a+ 61—2,  X  =  ix  for  a  perfect  solid,  a  =  a~i,  b=^bi—i, 

-^i  =  WaJ  -^2  =  ^^;  hence 

Wa  =  ^^(3«+^-4), 

^6  =  ^(^+3^1-4), 

and  the  equation  of  the  stress-ellipse 

X"  y^ 

7^.+  ^T7—, — 7 r,  =  I. 


P(^a+b-4y'^P{a+sK-4y 


§  59.   The   Rectangular  Pair  of  the   Involution   of  Stresses  in 

Nature. 

In  the  sections  corresponding  to  the  rectangular  pair  of  the 
involution  only  normal  stresses  are  acting  and  these  represent 
the  maximal  and  minimal  normal  stresses.  If  at  the  point  con- 
sidered we  advance  an  infinitesimal  amount  in  the  direction  of 
one  of  the  conjugate  rectangular  sections,  for  instance  that  for 
which  the  normal  stress  is  a  maximum,  we  can  at  this  place, 
infinitely  close  to  the  first,  again  construct  or  calculate  the  two 
rectangular  pairs  of  the  involution.     On  the  section  for  which 

^  Theorie  der  Elasticitat  fester  Korper,  Leipzig,  1862. 

^  Minchin:   Treatise  on  Statics,  Vol.  II,  p.  435,  fourth  edition. 


240 


PROJECTIVE   GEOMETRY. 


the  normal  stress  is  a  maximum  we  can  again  advance  an  infini- 
tesimal distance  and  then  construct  the  two  conjugate  normal 
sections,  etc.  In  this  manner  a  curve  is  obtained  along  which 
the  normal  stresses  have  their  maximal  values.  If  these  stresses 
are  tensions  which  are  greater  than  the  elastic  limit  of  the  material, 
then  the  material  will  rupture  along  this  curve  (maximal  tension 
curve).  In  a  similar  way  a  curve  may  be  drawn  through  the 
point  along  which  the  normal  stresses  have  their  minimal  values. 
If  the  involution  is  hyperbolic,  this  curve  is  a  maximal  compres- 
sion curve,  since  the  stresses  along  this  curve  are  maximal  with 
reference  to  the  compressive  stresses. 


Fig.  I02. 


This  case  of  a  hyperbolic  involution  is  shown  in  Fig.  102, 
which  shows  the  crevasses  of  Arapahoe  glacier.*    The  stream- 


^  From  a  drawing  by  Professor  N.  M.  Fenneman  in  an  article:  The  Arapahoe 
Glacier  in  IQ02,  Journal  of  Geology,  Vol.  X,  p.  841. 


APPLICATIONS  IN  MECHANICS. 


241 


lines  represent  the  curves  of  maximal  compression,  while  the 
crevasses  cutting  the  stream-lines  orthogonally  represent  the 
maximal  tension  curves.  The  case  of  an  elliptic  involution 
where  there  are  only  tensions  is  illustrated  by  the  cracks  whicH 
form  on  a  heavily  varnished  surface  or  in  a  mud-bed  which  is 
drying  up.  In  this  case  only  tensile  normal  stresses  act  on  the 
rectangular  pair.  One  is  a  maximum,  the  other  a  minimum. 
We  should  therefore  expect  that  the  maximal  tension  curves 
would  form  a  system  of  more  or  less  parallel  curves.  This, 
however,  does  not  occur,  as  is  seen  from  Fig.  103,  in  which  the 


Fig.  103. 

cracks  intersect  or  meet  orthogonally.  This  can  be  explained 
in  the  following  manner:  After  a  crack  has  formed,  the  maximal 
stress  and  strain  normal  to  the  crack  has  been  relieved,  so  that 
the  former  minimal  normal  tension  along  the  crack  now  becomes 
the  maximum.  Hence  the  next  rupture  will  be  orthogonal  to 
the  first  crack.  ^ 


^  See  my  article  on  this  subject  in  the  American  Mathematical  Monthly,  Vol. 
VII,  pp.  134,  135.  Further  examples  may  be  found  in  Ritters  Graphische  Statik, 
Vol.  I,  pp.  128-134. 


242  projective  geometry. 

Realization  of  Collineations  by  Linkages. 

§  60.   Introductory  Remarks. 

We  have  seen  that  coUineations  may  be  produced  analytically 
and  synthetically  by  different  methods.  In  what  follows  a  num- 
ber of  linkages  will  be  described  by  which  collineations  may  be 
realized  kinematically.  Linkages,  like  pantographs,  translators, 
etc.,  devised  for  some  special  purpose  have  been  known  for  a 
long  time.  The  history  of  linkages  in  connection  with  the  theory 
of  geometrical  transformations,  however,  dates  back  to  the  year 
1864,  when  Peaucellier  found  a  rigorous  solution  for  the  prob- 
lem to  describe  a  straight  line  by  a  link-motion.^  Since  that 
time  a  number  of  geometers,  among  whom  the  English  Sylvester, 
Hart,  Roberts,  Cayley,  and  Kempe  occupy  the  foremost  places, 
have  made  a  systematic  study  of  linkages  and  their  geometric 
properties  and  have  found  a  great  number  of  important  results. 
Among  these  investigations  probably  the  most  interesting  are 
those  of  Kempe  and  Koenigs.  The  first  proved  the  theorem 
that  it  is  always  possible  to  find  a  linkage  so  that  one  of  its  points 
describes  any  given  algebraic  curve.  Koenigs  generalized  this 
by  proving  that  every  algebraic  surface  and  curve  may  be  described 
by  a  linkage.  As  a  result  of  these  interesting  theorems  it  is  not 
difficult  to  prove  that  any  algebraic  transformation  between  any 
number  of  variables  may  be  realized  by  linkages.^  The  diffi- 
culty lies  in  the  actual  construction  of  such  linkages.  Recently 
Koenigs  has  invented  a  linkage  which  realizes  a  general  projec- 
tive transformation  in  a  plane.  ^ 

^  Nouvelles  Annales  de  Mathematiques,  2d  series,  Vol.  Ill  (1864),  p.  144. 

^  Koenigs:  Legons  de  Cinematique  (1897),  pp.  262-307.  See  also  Transac- 
tions of  the  American  Mathematical  Society,  Vol.  Ill,  pp.  493-498  (Oct.  1902), 
where  the  author  proves  that  any  number  of  algebraic  relations  between  n  complex 
variables  may  be  realized  by  a  plane  linkage. 

^  Comptes  Rendus,  Vol.  CXXXI,  p.  11 79.  The  different  cases  of  coUineation 
were  worked  out  by  Hermann  Emch  in  his  Master's  Thesis  at  the  University  of 
Colorado,  1902. 


APPLICATIONS  IN  MECHANICS.  243' 

To  have  a  definite  idea  about  the  character  of  the  plane 
linkages  to  be  considered  I  set  down  Koenigs's  definition: 

A  plane  linkage  (systeme  articule,  Gelenkwerk)  is  a  combina- 
tion 0}  plates  or  plane  figures  subject  to  remain  in  one  and  the 
same  plane,  among  which  a  certain  number  are  connected  to  each 
other  by  hinges  or  pivots  perpendicular  to  the  common  plane. 

In  this  definition  it  is  assumed  that  the  Hnks  move  by  each 
other  without  interference,  which  means  that  the  Hnks,  consid- 
ered as  material,  lie  in  a  series  of  close  parallel  planes. 

Every  linkage  is  constructed  in  such  a  manner  that  one  of 
its  pivots  is  fixed  and  represents  the  origin  O,  while  others  repre- 
sent the  algebraically  related  variables.  The  points  of  the  link- 
ages will  always  be  designated  by  the  same  letters  as  the  corre- 
sponding variables. 

Two  or  more  linkages  each  involving  two  variables  may  be 
combined  in  the  following  manner:  Suppose  L,  L^,  L^  .  .  .  ,  Ln 
are  linkages  realizing  the  transformations 

Let  the  origins  of  all  these  linkages  coincide;  attach  the  pivot 
Un  of  Lyi  to  the  pivot  Un  of  Ln-i',  attach  the  pivot  m„_i  of  I<„_i  to 
w„_i  of  Ln-2^  and  so  forth;  finally  the  pivot  u^  of  L^  to  u^  of  L. 
Then  the  point  il  of  L  evidently  realizes  the  compound  trans- 
formation 

Linkages  involving  more  than  two  variables  may  be  similarly 
combined. 

The  range  of  effectiveness  of  a  linkage  is,  of  course,  limited 
to  a  certain  finite  portion  of  the  plane.  This  range,  although 
in  some  cases  small,  always  exists. 


244  PROJECTIVE   GEOMETRY. 


§  6i.   Analytic  Formulation  of  the  Problem. 

We  shall  consider  only  the  most  important  cases  of  coUinea- 
tion.  A  great  number  of  special  cases  will  be  left  as  exercises 
for  the  student. 

The  most  important  cases  are  the  linear  transformation  and 
perspective.     A  hnear  transformation 

J  x^  =  ax-^by-\-c, 
^^^  \y^  =  dx+ey+j 

may  be  considered  as  made  up  of  four  subgroups:  (i)  the  two- 
termed  translation,  (2)  the  one-termed  rotation,  (3)  the  two- 
termed  dilation,-  (4)  the  one-termed  elation.  By  a  translation 
(p,  q)  and  a  rotation  (<^)  the  point  {x,  y)  is  changed  into  (x',  y'). 

(  o(/  =x  cos  (f>—y  sin  (f)+  p, 
^^  \  y' =x  sin  (j)  +y  cos  cf)-{-q. 

A  dilation 

'  x''=ax'. 


(3)  [  y/  =^y 

changes  {x/,  /)  into  (x",  y") : 

[  xf^  =  ax  cos  (j>—ay  sin  (f)-{-ap, 
^^\  \  y"  =^x  sin  4^-\-^y  cos  ^  +  /?g. 

Finally  by  the  elation 

X^  =  X"  -f  yl/', 


I    y    _  a;' 

we  get 

J  Xi  =  (a  cos  (;6+/^(5  sin  ^)x+ (>^j5  cos  ^-a  sin  <p)y-\-ap+k^q, 

I  3'i=/^  sin  </)•.%-+/?  cos  4>-y+l^q, 

*  Term  used  by  S.  Lie,  loc.  cit.     It  is  equivalent  with  dilatation,  p.  60. 


APPLICATIONS  IN  MECHANICS.  245 

which  by  properly  choosing  a,  /?,  X,  cf),  p,  q,  the  six  parameters, 
may  represent  any  hnear  transformation  of  (x,  y)  into  {x^,  y^). 
To  prove  this  let 

a  cos  4>+ X^  sin  4>  =  (i, 
X^  cos  0— a  sin  0  =  &, 

/?  sin  ^  =  (/, 

/?  cos  <?!)  =  e, 

/??  =  /, 

which  represent  six  equations  with  six  unknown  quantities  a,  /?, 
X,  ^,  p,  q.     Solving,  we  get 

d            c(d'- -he'-)-}  (ad+be)  f 

9  =  arctan— ,     p=- ri ,     q  = 


ae-bd  '     ^     Vd^+e^' 

Substituting  these  values  in  (2),  (3),  (5),  we  obtain  a  trans- 
lation with  rotation,  a  dilation,  and  an  elation  which  in  succession 
transform  {x,  y)  into  (.v',  y'),  {x\  y')  into  {xf' ,  y"),  and  finally 
{x",  y")  into  {x^,  y^)  in  such  a  manner  that  (x^,  y^  is  connected 
to  {x,  y)  by  the  linear  transformation 

\x^  =  ax+by+c, 
^'^  \y^  =  dx-{-ey+f. 

Applying  to  (x^,  y^)  the  perspective 


<8) 


X'  = 


d,x,+  e,y,  +  fi' 


y-      ' 


^i^i+«i3'i+/i' 


246  PROJECTIVE  GEOMETRY. 

we  have 


(9) 


ax-\-hy-{-c 
X  = 


y- 


(d^ai-  e^d)x+  {dJ)^e^e)y-\-}^* 

dx-\-ey+  f 
{d^a+  e^d)x-\-  {dyb-^e^e)y-\-}^' 


which  may  represent  any  projective  transformation.  The  linear 
transformation  is  a  six-termed  and  the  perspective  a  three- 
termed  group,  so  that  their  combination  (9)  contains  nine  param- 
eters, although  the  general  projective  group  is  eight-termed. 
This  is  due  to  the  fact  that  both  the  linear  transformation  and 
the  perspective  contain  the  one-termed  group  of  similitudes  as  a 
subgroup.  These  considerations,  which  may  be  found  in  a  little 
different  form  in  §  19,  have  been  repeated  here  for  a  clearer  under- 
standing of  what  follows. 

We  shall  now  proceed  to  describe  linkages  realizing  the  trans- 
formations in  question.  Theoretically  only  such  linkages  should 
be  admissible  in  which  a  link  joins  two  and  only  two  points. 
In  other  words,  no  three  points  in  a  straight  lire  should  be  ad- 
mitted a  priori.  It  is,  however,  very  useful  for  practical  pur- 
poses to  make  this  last  assumption.  For  some  transformations 
we  shall  construct  more  than  one  linkage  in  order  to  show  the 
advantage  which  one  or  the  other  may  have. 

Combining  the  linkages  involved  in  the  linear  transforma- 
tion and  in  perspective  according  to  the  scheme  explained  in 
the  last  part  of  §  60,  a  compound  linkage  for  a  general  colHnea- 
tion  is  obtained. 

§  62.   Peaucellier's  Inversor. 

In  our  particular  investigation  of  link-motions  the  problem 
to  draw  a  straight  hne  theoretically  correct  is  of  the  greatest  im- 
portance. This  can  be  done  by  Peaucelher's  inversor  (loc.  cit.) 
or  by  Hart's  linkage  (Koenigs's  Cinematique,  p.  267).  Peau- 
celher's inversor  is  of  greater  principal  value  and  will  be  described 
here. 


APPLICATIONS  IN  MECHANICS. 


H7 


It  consists  in  the  first  place  of  a  rhombus  ABPP'  and  two 
equal  links  AO  and  BO.     In  all  these  points  the  links  are  joined 


Fig.  104. 

by  pivots,  Fig.  104.  Designating  OA=OBhy  a,AP  =  PB  =  BP'  = 
P'A=b,AA'  =  c,weh3iYeOA'  =  h{OP+OP'),A'P  =  ^(OP'-OP); 
i(OP+OPy  =  a'-c';  c^  =  ¥-\{OP'-OP)\  or  \{pP-^OP'y- 
l{0F-OPy  =  a'-h\  or  finally 

OP-OP'  =  a'-h\ 

Hence,  if  O  is  kept  fixed,  the  points  P  and  P'  are  inverse 
with  respect  to  a  circle  having  O  as  a  center  and  ^a^—h'^  as  a 
radius.  If  now  P  describes  a  circle  with  M  as  a  center  and  OM  =  r 
as  a  radius,  we  have  OP-OP' =  OT-OQ,  or  OP/OT  =  OQ/OP'; 
consequently  aOPT  fOA  OQP';  and  since  Z  OPT  =  90°,  also 
Z  OQP'  will  be  a  right  angle.  Hence,  when  P  describes  said 
circle,  P'  ivill  describe  a  straight  line  perpendicular  to  the  direction 
of  OM. 

For  the  limiting  position  OSRS'  of  the  inversor  we  have 


or,  smce 


QS'=^VOS''-OQ\ 


OS'^={a+by    and    0Q  = 


a'-¥ 


2r 


QS'  =  ~V{a+by4r'-  {a^-  b^. 


248  PROJECTIVE  GEOMETRY. 

Of  course  the  lengths  a,  b,  r  must  be  chosen  in  such  a  manner 

a—  b 

that  the  hnkage  is  movable.     The  conditions  are  r> ,  f ol- 

2 

lowing  from  (a+&)^4r^— (a-— Z)^)^>o,  and,  for  the  case  that  the 
straight  line  shall  not  cut  the  circle  of  inversion  r<  h\^a^~  P. 

Ex.  I.  Show  that  when  M,  P,  and  A  are  in  a  straight  line, 
AP'±S'Q. 

Ex.  2.  If  the  whole  linkage  is  in  a  vertical  plane  and  OM 
vertical,  the  linkage  remains  in  equilibrium  under  the  action  of 
any  weight  suspended  at  P'. 

Ex.  3.  If  a  and  b  are  given,  what  value  must  r  have  to  make 
QS'  a  maximum  ? 


§  63.   Pantographs. 

I.  Inversor  Pantograph. 

By  means  of  Peaucellier's  cell  ABPQO,  a  part  of  the  inversor, 


Fig.  105. 


we  can  locate  for  every  point  P  sl  point  Q,  so  that  OP ■  OQ^a^—  P. 
Applying  another  cell,  A, B.Q'P'O,,  for  which  0,Q'  ■  0,F=a,'-  b,\ 


APPLICATIONS  IN   MECHANICS. 


249 


and  letting  O^  coincide  with  O,  and  Q  with  Q' ,  Fig.  105,  then 
O^Q'  =  0Q'  J  and  by  division 

OP     a^-b^ 
OP'  ~  a,'-  6,2- 

Hence,   when  P  describes  a  figure,  P^  will  describe  a  similar 
figure.     Choosing  O  as  the  origin  of  Cartesian  coordinates  and 

designating  the  constant  ratio  — 5 — 7-3  by  k,  then  when  O  is  fixed, 

the  linkage  of  Fig.  105  will  realize  the  transformation  of  P{x,  y) 
intoP'(x', /): 

0(/  =  KX,     y'  =  Ky. 

2.  Sylvester^ s  Pantograph.^ 

Take  any  two  similar  triangles,  Fig.   106,  OAA'  and  APB 
pivoted   at  A,   with    IA'0A=  iBAP   and     /.A'AO=  IBPA. 


Now  IQA'P'=IA'AB,  ZQA'0=  ZA'OA+  ZA'AO;  hence 
IQA'P'+  ZQA'0=  Z.A'AB+  IBAP+  ZA'AO,  or  ZOA'P'  = 
Z.OAP.     But  there  is  also 

OA      AP     AP 


OA'    AB    A'P'' 


^  Nature,  1875,  p.  i( 


250  PROJECTIVE   GEOMETRY. 

OA      OP        ,       ,^„ 
hence   aO^'P'^aO^P.     From  this  Qj,  =  Qp}  and  IA0P  = 

ZA'OP',  hence  also   ZPOP'=  lAOA'  and   aPOP'w  a^O^'. 

Consequently  when  P  describes  a  figure  and  O  remains  fixed, 

P'  will  describe  a  similar  figure.     The  ratio  of  simihtude  between 

the  figures  traced  by  P  and  P'  is  OA/OA'.     Turning  the  figure 

traced  by  P'  negatively  through  an  angle  =  AA'OA  it  will  come 

similarly  situated  with  the  figure  traced  by  P  with  respect  to 

the  center  O.     Designating  by  0  the  angle  A'OA,  and  by  p  the 

OA' 
ratio  T^rj-,  Sylvester's  pantograph  will  realize  the  combined  groups 

of  rotation  and  similitude  between  P{x,  y)  and  P'{x',  /): 

od  =  p{x  cos  (})—y  sin  ^), 
/  =  jo(x  sin  <j!>  +  }' cos  0). 

This  becomes  a  pure  rotation  when  p=i',  i.e.,  OA  =0A'. 

The  arrangement  of  this  linkage  is  a  little  different  from 
Sylvester's  original  pantograph,  but  does  not  essentially  differ 
from  it. 

3.  P.  Schemer's  Pantograph  (1631). 

Scheiner's  or  the  ordinary  pantograph  appears  in  the  market 
under  many  different  forms.  One  of  the  simplest  is  illustrated 
in  Fig.   107.     Two  equal  sets  of  links  PQ,  PR  and  CP\  CO 


pivoted  at  P  and  C  are  placed  in  such  a  manner  that  P  is  in  a 
straight  line  with  O  and  P',  andP(3liCP^  PR\\CO.  In  this 
position  pivots  are  also  placed  where  PR  and  CP',  and  PQ  and 
CO  meet.  From  the  figure  it  appears  at  once  that  when  O  is 
fixed    and  P  describes  a  figure,  then  P'  will  describe  a  figure 


APPLICATIONS  IN  MECHANICS.  251 

similar  to  and  similarly  situated  with  the  first.  The  linear  ratio 
between  the  two  figures  is  OP /OP'.  To  make  different  values 
for  this  ratio  possible  the  hnks  may  be  divided  into  equal  parts, 
as  shown  in  Fig.  107.  Wishing  to  enlarge  a  figure  in  the  linear 
ratio  4  : 3,  set  the  pivots  where  PR  and  CP' ,  and  PQ  and  CO  meet 
at  the  marks  6,  so  that  OP'  /OP  =  8:6  =  4:3.  In  a  similar  manner' 
arrangements  for  any  other  ratio  may  be  made  by  properly 
dividing  the  links. 

Although  Scheiner's  pantograph  is  the  simplest  of  all  panto- 
graphs and  consequently  exclusively  used  for  practical  purposes, 
it  has  the  theoretical  disadvantage  of  not  being  a  pure  linkage. 
Indeed,  in  Fig.  107  it  is  assumed  that  a  link  joins  three  given 
points  in  a  straight  line.  The  first  two  pantographs  described 
are  pure  linkages. 


§  64.    Rotator  and  its  Combinations.^ 

I.  To  realize  a  rotation  through  an  angle  ^  of  a  point  P{x,  y) 
into  P'{x' ,  y'),  so  that 

oc' =x  cos  (p—y  sin  (f), 
y'  =x  sin  0+7  cos  0, 

Sylvester's  pantograph  in  the  case  ,0=1  may  be  used.     Another 

linkage  for  the  same  purpose,  Fig.  108,  is  obtained  by  taking 

two    isosceles  triangles    OAC    and    OBB 

pivoted   at    O,   the    coordinate-origin,  with 

l.AOC=  IB0B  =  <^  and  ^0  =  C0=50  = 

DO.     Attaching  the  links  PB  =  PC,P'A  = 

P'D,  pivoted  at  P  and  P'  respectively,  and 

all  equal  to  AO,  two  equal  rhombs  OBPC 

and  OAP'D  are  obtained.     Hence  lAOP  = 

I  BOP',  ZPOB=  ZP'OA,  and  ZBOP'  + 

ZPOB=  ZAOP+  ICOP,  or 

IP0P'=  lAOC  =  cl>. 

^  A  paper  on  this  Knkage  and  its  applications  was  presented  to  the  Am.  Math. 
Soc.  in  Chicago,  Sept.  1902,  and  was  pubHshed  in  Vol.  I  of  The  University  oj 
Colorado  Studies,  April  1903.  See  also  Transactions  of  the  Am.  Math.  Soc, 
Vol.  Ill,  No.  4,  pp.  493-498,  Oct.  1902. 


Fig.   108. 


252 


PROJECTIVE   GEOMETRY. 


Furthermore,  P'0=PO.     The  hnkage  of  Fig.  io8  can  therefore 
be  used  to  perform  the  proposed  rotation, 

2.  The  foregoing  hnkage  may  be  used  for  various  purposes. 
In  the  first  place  when  O  is  not  fixed,  we  have  in  the  three  pivots 
P,  O,  P'  a  variable  isosceles  triangle  which  in  all  its  deforma- 
tions remains  similar  to  some  original  size.  When  O  is  fixed  and 
P  describes  a  straight  line  or  a  circle,  P'  also  describes  a  straight 
line  or  a  circle  respectively.  Making  9!)  =  90°,  and  taking  two 
equal  rotators  with  the  points  P  and  P'  attached,  a  variable 
square  is  obtained. 


65.   Translators. 


One  of  the  simplest  devices  for  translation  is  that  of  Kempe.^ 
It  consists  of  three  parallel  equal  links  AD,  BC,  PP'  which  are 

connected  by  CD  =  \\AB  and  CP'  = 
\\BP.  Letting  A  coincide  with  the 
coordinate-origin  and  designating  the 
coordinates  of  D  by  a  and  h,  then  for 
the  coordinates  {x,  y)  of  P  and  (x^,  /) 
of  P'  we  have  from  Fig.  109 
Of/  =  x-^a, 
Fig.  109.      ■  y'-y+h. 

A  translator  which  is  more  general 
is  obtained  from  a  linkage  which  was  originally  invented  to  per- 
form the  addition  of  any  complex  variables. ^  It  consists  of  12 
links,  Fig.  no,  of  which  OF  =  \\CE  =  \\BD-  FP' =  \\EG==\\DP■ 
OC=\\FE=\\  P'G ;  CB  =  \\ED  =  \\ GP.  It  is  evident  that  OBPP' 
will  always  be  a  parallelogram  no  matter  how  the  hnkage  may 
be  deformed.  Hence,  keeping  B  and  O  fixed,  P'  represents  in 
every  position  of  the  linkage  a  translation  of  P  equal  to  BO  and 
in  the  direction  of  BO. 

^  How  to  Draw  a  Straight  Line. 
^  Transactions,  loc.  cit. 


APPLICATIONS  IN  MECHANICS. 


253 


§  66.   Linear  Transformation. 

I.  By  a  combination  of  rotator  and  translator  it  is  possible 
to  realize  a  general  motion  in  the  plane.     According  to   §  61 


Fig.  no. 

the  next  group  in  making  up  a  linear  transformation  is  the  dila- 
tion 

oc"=ao(f. 

y"=^y'. 

The  linkage  for  this  transformation  is  shown  in  Fig.  iii.^ 
Let  OA=x'.  By  a  Scheiner  pantograph  (which  we  choose  for 
the  sake  of  simplicity),  in  which  OB/OA=a  and  consisting  of 
the  links  OC,  CB,  DA  and  AE,  a,  point  B  is  realized  for  which 
B0=o</'  =ax'.  One  of  the  points  A  and  B  is  kept  on  the  :r-axis 
by  a  Peaucellier  inversor.  To  A  and  B  attach  a  translator 
ABIP'VLMNH.     Produce  the  link  NB  arbitrarily  to  R  and 


^  This  is  essentially  the  arrangement  of  Hermann  Emch  in  his  thesis,  loc.  dt. 


254 


PROJECTIVE   GEOMETRY. 


complete  the  rhombus  BRST.  To  BR  and  BT  attach  at  B  the 
equal  right-angled  triangles  RBF  and  TBG,  so  that  BF=BG. 
Complete  the  rhombus  BFGP" .  From  the  figure  follows  easily 
that  IFBG=  ZRBT  and  that  P"B1.BS,  LOB. 

Now  use  FB  and  FP"  as  links  of  a  second  Scheiner  panto- 
graph, and  attach  the  links  IK  and  /Z7  in  such  a  manner  that 


Fig.  III. 

BF:IK  =  F'F:IU=BP"  .IP",  and  P"B/BI=^.  The  point  / 
is  now  colhnear  with  P"  and  B,  and  as  P'A'^IB  it  follows  that 
P'A  _L  OA .  flaking  P'A  =  /,  we  find  IB  =  P'A  =  y' ;  P"B  = 
^■IB=^y' .     The  coordinates  of  P"  are  therefore 

BO^x^^ax', 
pf'B  =  y"=l3y, 


and  we  have  constructed  a  linkage  realizing  dilation. 

2.  The  last  group  to  be  considered  is  the  one-termed  elation. 
Take  two  rhombs  AEP"F  and  ACBD  with  the  common  joint  or 
pivot  A  ;  join  E  and  C,  and  F  and  D  by  two  equal  links  EC  =  FD, 


APPLICATIONS  IN  MECHANICS. 


^SS 


so  that  lEAC- 


ZFAD=-. 

2 


From  plane  geometry  there  fol- 


lows easily  ZEAF=  ICAD;  i.e.,  that  the  two  rhombs  are  similar; 
further,  that  P"A  A.  AB,  no  matter  how  the  linkage  may  be 
distorted.  This  linkage  realizes,  therefore,  a  variable  right  tri- 
angle P"AB  whose  angles  are  constant.  Joining  in  a  symmetrical 
manner  the  rhomb  BHPfi^AEF'F  to  the  previous  hnkage 
(CG  =  CE,  HD  =  FD),  a  variable  rectangle  ABP"P^  is  obtained 
whose  sides  have  a  constant  ratio,  Fig.  112. 


0- 


Fig.  112. 


This  linkage  may  be  used  to  solve  mechanically  two  interest- 
ing cases  of  collineations  in  a  plane.  If  by  two  Peaucellier 
Inversors  the  points  A  and  B  are  forced  to  describe  the  same 


256  PROJECTIVE  GEOMETRY. 

Straight  line  s,  in  which  an  arbitrary  point  is  taken  as  the  origin 

of  a  Cartesian  coordinate  system,  and  5  itself  is  assumed  as  the 

AB 
:v-axis,  we   have,  since  "7^/7"^''^  (constant),  for  the  coordinates 

i^i,  ^'i  of  Pi  in  terms  of  those  of  P"{x",  y")'. 

x^  =  x"  -^  my" , 

which  represents  the  required  elation. 

If  the  rhombus  BGHP^  and  the  links  CG  and  DH  are  attached 
below  s,  so  that  the  point  Py  will  fall  on  P2,  then  the  coordinates 
of  P2  are 

X2  =  o<f'  -\-my" , 

which  represents  oblique  axial  symmetry.  Combining  the  link- 
ages for  rotation,  translation,  dilation,  and  elation  as  explained 
in  §  60,  a  linkage  for  the  linear  transformation  is  obtained. 

Ex.  I.  Construct  a  linkage  for  the  transformation  (oblique 
axial  symmetry): 

oc' =x-{-my, 

y'  =  -y. 

Ex.  2.  Construct  a  linkage  for  the  transformation  (orthogonal 
axial  symmetry) : 

y'  =  -y, 
Ex.  3.  Construct  a  linkage  for  the  special  dilation: 
x'  =ax^ 

y=y. 


APPLICATIONS  IN  MECHANICS.  257 

Ex.  4.  Construct  a  linkage  for  the  central  symmetry: 

y--y- 

Ex.  5.  Draw  the  combined  linkage  for  a  general  linear  trans- 
formation. 

Ex.  6.  Determine  the  ranges,  or  hmits  of  the  areas,  covered 
by  Sylvester's  pantograph,  the  rotator,  the  translators,  and 
Scheiner's  pantograph  as  used  in  the  linkage  for  dilation. 

§  67.    Perspective. 

1.  Mechanisms  by  which  the  perspective  of  any  plane  figure 
may  be  drawn  are  known  in  various  forms.  One  that  is  in  prac- 
tical use  is  the  " perspectivograph "  invented  by  H.  Ritter.^  In 
this  mechanism  pivots  are  kept  on  given  straight  lines  by  grooves 
so  that  it  combines  link-  and  sliding-motions.  Another  "per- 
spectivograph" in  which  two  elhpses  are  used  and  which  also 
combines  link-  and  sHding-motions  was  described  by  the  author 
some  years  ago.^  Probably  the  most  important  linkage-realizing 
perspective  has  been  invented  by  Koenigs,^  and,  as  it  does  not 
use  slide-motion,  will  be  described  here.  We  must,  however,  first 
describe  Kempe's  reversor  which  Koenigs  uses  as  an  auxiliary 
linkage. 

2.  Kempe's  Reversor. 

In  Fig.  113  consider  the  linkage  OBCD  in  which  OB  and  CD 
are  equal  and  cross  each  other,  and  also  OD^BC.  Desighating 
the  variable  point  of  intersection  of  OB  and  CD  by  X,  this  linkage, 
which  is  called  counter-parallelogram,  has  the  property  that  for 
any  deformation  A ^0Z)=  A Z)C5;  aOXD=aCXB.  On  DC 
choose  a  pivot  E  in  such  a  manner  that  DE:DO=DO:DC,  so 
that  the  triangles  OCD  and  EOD  are  similar.     Then  with  OD 

^  See  Geometrische  Transjormationen  by  Dr.  K.  Doehxemann,  pp.   199-204, 
Leipzig,  1902. 

^  The  Industrialist,  Vol.  XXV,  pp.  237-240,  Manhattan,  Kansas,  1899. 
^  Comptes  Rendus,  Vol.  CXXXI,  p.  11 79. 


258 


PROJECTIVE  GEOMETRY. 


and  ED  as  given  links  complete  the  counter-parallelogram  ODEF, 
in  which  FE  =  DO,  FO=DE.  Thus  aEOD=aOEF  and 
similar  to   aOCD=  aCOB,     Hence,  in  every  deformation,   the 


Fig.  113. 

counter-parallelograms  QBCD  and  ODEF  are  similar,  and  as 
A  BOD  =  A  DCS,  it  also  follows  that  a  DOF  =  A  FED.  By  means 
of  this  reversor  it  is  possible  to  keep  two  links  BO  and  FO  at 
equal  angles  from  a  given  link  DO.  By  a  similar  construction 
two  other  links,  B'O  and  F'O,  symmetrical  to  DO  may  be  attached, 
and  it  is  clear  that  their  motion  is  otherwise  independent  of  that 
of  BO  and  FO.  We  have  therefore  a  linkage  in  which  in  every 
deformation 

IBOB'^IFOF'. 

Kempe's  reversor  may  be  extended  to  realize  any  number  of 

equal  angles,  ABOD=  Z.DOF=  ZFOH= For  the  details 

of  this  we  refer  to  Koenigs's  Kinematics,  loc.  cit. 


KoENiGs'  Perspectivograph. 

3.  Introducing  polar  coordinates,  x  =  rcos  d,  y  =  rs'm  d,  in  the 
formulas  for  a  perspective  transformation 


j/  = 


(I) 


i 


dx-\-ey+f 

y 


y= 


APPLICATIONS  IN  MECHANICS. 


259 


we  get,  since 


:arc  tan  (  -  j  =arc  tan  (— ,  ,, 


(2) 
and 

(3) 


/  = 


r{d  cos  d+e  sin  d)-\-f' 


-j  =  d  cos  d+e  sin  (9+-. 
/  r 


Putting  ^=-^  sine?!),   e  =  --  cos(}^,  so  that  a=^^j^=  and 


</»  =  tan~M  -  j,  (3)  becomes 
<4) 


'---^=--sin(^+c^). 
r     r         a 


Now  take  two  Peaucellier  inversors  OABQP  and  OCDQ'P', 
Pig.   114,  and  by  means  of  Kempe's  reversor  as  described  in 


Fig.  114. 


Fig.  113  keep  lAOC=  IBOD.    This  can  be  done  by  properly 
choosing  B' ,  F,  B,  F  of  Fig.  113  on  BO,  AO,  DO,  CO  of  Fig.  114, 


26o  PROJECTIVE   GEOMETRY. 

respectively.     Let  OP  =  r,  OP'=r'\    {i  and  /  the  squares  of  the 

radii  of  the  circles  of  inversion  of  the  two  inversors;  OQ  =  p, 

I        p 
OQ'  =  p'.       Now     OP-OQ  =  fi,     or     r-p  =  /i,     hence  — = — ; 

I  p' 

OP'  •  OQ'  =  /,  or  r'  ■  p'  =  jj! ,  hence  —  =  — ^ .     So  far,  no  particular 

values  are  assigned  to  p.  and  p.',  so  that  we  can  choose  p  =  j-  p!] 

f        P 

—  =  —y.     Equation  (4)  now  becomes 

(5)  ^'-^=— ^  sin(^+9S). 

To  the  two  inversors  attach  a  Kempe  translator  OO'EFRQ, 
where  O'  is  on  the  j-axis,  and  join  R  to  Q'  by  the  link  RQ'  =RQ. 
Let   Z.POX  =  d-^<t>\  then   IQRQ' =  2 {d +<}>),  and  QQ'  =  p'-p  = 

2RQ-sm  {d-\-(f>).  Hence,  taking 00' =  i?(2  =  --  —  =  ~Vd^+e\ 

the  points  P'  and  P  realize  the  proposed  perspective  transforma- 
sion,  since  the  linkage  satisfies  all  conditions  of  equations  (i)  or 
(2),  or  their  equivalent  (5). 

Ex.  I.  Modify  the  linkage  so  that  P  and  P'  describe  similar 
figures. 

Ex.  2.  Investigate  the  cases  /=i,  and  /=o. 


INDEX. 


PAGE 

Affinity 53,  77 

Analytic  expression  for  a  Steinerian  transformation 187 

expression  for  tangent  and  polar 118 

formulation  of  the  problem  of  linkages  for  coUineations 244 

representation  of  central  projection 49 

Angle  mirror 171 

Anharmonic  ratio 5 

An  optical  problem 167 

Apollonius 92 

Applications;  of  perspective 86 

in  mechanics 217 

Asymptotes  of  conies 104 

Axes  of  conies 104 

Axis  of  perspective 30 

Bicircular  cubics 190 

Bobillier 136 

Brianchon 133,  175 

Brianchon  point 135 

Brianchon's  theorem 134 

Cajori 179 

Casey 80 

Cayley 42 

Cayley's  theorem 137 

Center;  of  a  conic 103 

of  perspective 30 

Central  projection.    45,  86,  105,  146 

special  cases  of ; 51 

Chasles 59,  92,  222 

Circular  cubics 204 

261 


262  INDEX. 

PAGE 

Circular  points  at  infinity 20,  41 

Classification  of  cubics 195 

Clebsch 203 

Collineation 45;  59 

Complete  quadrilateral,  the , 26 

Conies  as  intersections  of  right  cones 149 

Conic  sections 36,  92 

Conies  in  mechanical  drawing  and  perspective 140 

Conjugate  diameters 103 

Construction;  of  a  parabola,  etc 143 

of  an  equilateral  hyperbola,  etc 145 

of  conies 146 

of  foci  independent  of  central  projection 107 

of  projective  pencils  and  ranges 30 

Continuous  groups  of  projective  transformations 66 

Cremona 16,  166,  168 

Cubic;  with  oval  and  serpentine 19S,  209 

the  simple 19S,  211 

with  an  isolated  point 196,  211 

nodal 196,  212 

cuspidal 196,  213 

Cubics 189 

Curve  of  the  second  order 23 

Curves  of  the  second  order  and  class 24,  93 

Curves  of  the  third  order 189 

Czuber 207 

Desargues Hj  45;  ^74 

Diameters  of  conies 103 

Dilatation 60 

Directrix  of  a  conic no 

Discriminant 99 

Discussion  of  collineation 63 

Distance  circle 45 

Disteli 185,  203 

Doehlemann 146,  257 

Double  points  of  the  transformation 9 

Duality ■ 68 

Elation •  ■  56,  79 

EUipse 99.  104,  140 

Emch,  Hermann 242,  253 


INDEX.  263 

PAGK 

Enriques iii 

Equation  of  a  conic  in  line  coordinates 122 

Equilateral  hyperbola 145 

Euclid 5 

Exercises  and  problems 25,  34,  57,  70,  90 

Existence  of  ellipse,  hyperbola,  parabola,  and  their  foci 104 

Feuerbach  circle 1 79 

Fiedler iii,  25,  45,  137,  175 

Focal  properties  of  conies 109 

Foci;  of  a  conic 104 

construction  of 107 

Funicular  polygon 217,  219 

General  collineation 59 

General  construction  of  projective  pencils  and  ranges 30 

General  reciprocal  transformation 126 

Geometric  determination  and  discussion  of  collineation 63 

Geometric  quantities  and  their  signs i 

Geometry  of  stresses  in  a  plane 223 

Gergonne 68 

Groups  of  transformation 11,  66 

Harmonic  ratio 12 

Hart 242 

Hilbert 136 

Homologous  triangles 80 

Hyperbola 99,  145 

Identity  of  curves  of  the  second  order  and  class,  and  conies /.  .  .  93 

Intersection  of  conic  and  straight  Une -.  161 

Invariant  elements 68 

Inversor  pantograph 248 

Involution 11,51 

Involution;  of  conjugate  diameters 104 

of  poles  and  polars 41 

of  conjugate  sections  and  stresses 224 

of  stresses  in  nature 239 

of  the  pencil  U+W^  —  o 172 

Involutoric  pencils 18 

Involutoric  transformations 12- 


264  INDEX. 

PAGE 

Joachimsthal 26,  172 

Kempe 242 

Kempe's  reversor 257 

Kirkmann's  theorem 137 

Koenigs 242 

Koenigs'  perspectivograph 258 

Kotter 179 

Lagrange i 

Lame 22 

Lambert 45 

Laurent 2 

Leudesdorf 80 

Levy 229 

Lie II,  57,  68 

Linear  deformation 236 

Linear  transformation ; . . .  61,  244 

of  a  curve  of  the  second  order 94 

Maclamin's  theorem 35 

McCormack i 

Mechanical  drawing 140 

Menachmus 92 

Metric  properties  of  the  involution  of  stresses 229 

Minchin 222 

Moebius 3.  6,  7,  S3 

Monge 45>  149 

Newson. 34,  68 

Newton 196 

Newton's  theorem 35 

Oblique  axial  symmetry 56 

Optical  problem 167 

Orthogonal  axial  symmetry 56 

Orthographic  projection 73 

Osculating  circle  of  a  conic 159 

Pantograph;  Inversor,  Sylvester's 248,  249 

Scheiner's 250 

Pappus 16,  lie 


INDEX.  265 


PAGE 


Parabola 99,  104,  143 

campanifonnis  cum  ovali 195 

pura 195 

puncta 196 

nodata 196 

cuspidata 196 

Parallel  projection  of  conies 146 

Pascal 133 

Pascal  line 135 

Peaucellier 242 

Peaucellier's  inversor 246 

Pencils  and  ranges  of  conies 172 

Pencils  of  rays 15,  18 

Perspective 257 

Perspective;  between  any  two  given  conies 151 

of  a  square 86 

of  a  circle 88 

Perspective  pencils  and  ranges 29 

Perspectivograph,  Koenigs 258 

Picard 71 

Plucker 68 

Poincare 5 

Polar  and  tangent 1 1.3 

Polar  involution ;  of  the  circle 38 

of  conies 102 

Polar  reciprocity 131 

Polars  of  a  pencil,  of  conies 172 

Polar  systems-. 123 

Pole,  polar  of  a  circle 38 

Poncelet 7,  36,  45,  68,  92,  149,  175 

Poncelet's  principle  of  continuity 149 

Poncelet's  problem 164 

Poudra 45 

Principle  of  dua.lity 68 

Problem;  in  graphic  statics 219 

in  optics 167 

Problems  of  the  second  order 161 

Products  of  pencils  and  ranges  of  conies 180 

Products  of  projective  pencils  and  ranges 22 

Projection;  central 45,  86 

orthographic 73 

Projective  groups  of  transformation 11 


266  INDEX. 

PAGE 

Projective  properties  of  the  circle 35 

ranges  and  pencils 5,  15,  30 

theorems,  statical  proofs  of 220 

transformations  of  the  plane 59 

transformations  of  the  points  of  a  straight  line 5 

Quadrilateral 26 

Quadruple,  Steinerian 203 

Rabattement 77 

Realization  of  collmeations  by  linkages  242 

Reciprocal  polars 123 

Reciprocal  transformation 123,  126 

Rectangular  pair  of  an  involution 19 

Reye 5>  16,  92 

Reve's  configuration 86 

Ritter,  H 257 

Ritter,  W 228,  229 

Roberts 242 

Rotation 59 

Rotator 251 

Salmon 25,  135 

Scheiner's  pantograph 250 

Self-polar  triangle 42,  103 

Similitude 52 

Special  cases  of  central  projection 51 

Special  constructions  of  conies  by  central  projection  and  parallel  projection  146 

Statical  proofs  ol  some  projective  theorems 220 

Steiner 5,  22,  92,  185,  203 

Steinerian  quadruple 203 

Steinerian  transformation 185 

Steiner's  theorem 137 

Stress  ellipse 229 

Stresses  in  a  plane 223 

Sylvester 242 

Sylvester's  pantograph 249 

Symmetry 56 

Tangent  and  polar 118 

Tangents  from  a  point  to  a  conic 163 

Taylor 45 


INDEX.  267 

PAGE 

Theorems  of  Pascal  and  Brianchon 133 

Theory  of  conies 92 

Theory  of  reciprocal  polars 123 

Traces;  of  a  Une 75 

of  a  plane 76 

Transformation;  hyperbolic,  elliptic,  parabolic 9,  13 

projective 5,  59 

linear 61,  244 

groups  of 1 1,  66 

Translation 59 

Translators 252 

Various  methods  of  generating  a  circular  cubic. 204 

Veronese iii 

Von  Staudt iii,  5,  92 

Wiener iii,  45 

Wurf 5 


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Goodrich's  Economic  Disposal  of  Towns'  Refuse 8vo,  3  50 

Gore's  Elements  of  Geodesy , 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy. §vo,  3  00 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

5 


Howe's  Retaining  Walls  for  Earth r2mo,  i  25 

Johnson's  (J.  B.)  Theory  and  Practice  of  Surveying Small  8vo,  4  00 

Johnson's  (L.  J.)  Statics  hy  Algebraic  and  Graphic  Methods 8vo,  2  00 

Laplace's  Philosophical  Essay  on  Probabilities.     (Truscott  and  Emory.)- i2mo,  2  00 

Mahan's  Treatise  on  Civil  Engineering.     (1873.)     (Wood.) 8vo,  5  00 

*  Descriptive  Geometry 8vo,  1  50 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  2  50 

Elements  of  Sanitary  Engineering 8vo,  2  00 

Merriman  and  Brooks's  Handbook  for  Surveyors i6mo,  morocco,  2  00 

Nugent's  Plane  Surveying 8vo,  3  50 

Ogden's  Sewer  Design i2mo,  2  00 

Patton's  Treatise  on  Civil  Engineering 8vo  half  leather,  7  50 

Reed's  Topographical  Drawing  and  Sketching 4to,  s  00 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage , 8vo,  3  50 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry 8vo,  i  50 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) Bvo,  2  50 

Sondericker's  Graphic  Statics,  with  Applications  to  Trusses,  Beams,  and  Arches. 

8vo,  2  00 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  00 

*  Trautwine's  Civil  Engineer's  Pocket-book i6mo,  morocco,  5  00 

Wait's  Engineering  and  Archi'ectural  Jurisprudence 8vo,  6  00 

Sheep,  6  50 
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tecture  8vo,  5  00 

Sheep,  5  50 

Law  of  Contracts 8vo,  3  00 

•  Warren's  Stereotomy — Problems  in  Stone-cutting 8vo,  2  50 

Webb's  Problems  in  the  Use  and  Adjustment  of  Engineering  Instruments. 

■  i6mo,  morocco,  1  25 

*  Wheeler  s  Elementary  Course  of  Civil  Engineering 8vo,  4  00 

Wilson's  Topographic  Surveying 8vo,  3  50 

BRIDGES  AND  ROOFS. 

Boiler's  Practical  Treatise  on  the  Construction  of  Iron  Highway  Bridges.  .8vo,  2  00 

*  Thames  River  Bridge 4to,  paper,  5  00 

Burr's  Course  on  the  Stresses  in  Bridges  and  Roof  Trusses,  Arched  Ribs,  and 

Suspension  Bridges 8vo,  3  50 

Burr  and  Falk's  Influence  Lines  for  Bridge  and  Roof  Computations.  .  .  .8vo, 

Du  Bois's  Mechanics  of  Engineering.     Vol.  II Small  4to, 

Poster's  Treatise  on  Wooden  Trestle  Bridges 4to, 

Powler's  Ordinary  Foundations 8vo, 

Greene's  Roof  Trusses 8vo, 

Bridge  Trusses 8vo, 

Arches  in  Wood,  Iron,  and  Stone 8vo, 

Howe's  Treatise  on  Arches 8vo,  4  00 

Design  of  Simple  Roof-trusses  in  Wood  and  Steel 8vo,  2  00 

Johnson,  Bryan,  and  Turneaure's  Theory  and  Practice  in  the  Designing  of 

Modern  Framed  Structures Small  4to,  10  00 

Merriman  and  Jacoby's  Test-book  on  Roofs  and  Bridges : 

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Part  II.     Graphic  Statics 8vo,  2  50 

Part  III.     Bridge  Design 8vo,  2  50 

Part  IV.     Higher  Structures 8vo,  2  50 

Morison's  Memphis  Bridge .^ 4to,  10  00 

WaddeU's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers.  .  i6mo,  jnorocco,  3  00 

Specifications  for  Steel  Bridges lamo,  1  25 

Wood's  Treatise  on  the  Theory  of  the  Construction  of  Bridges  and  Roofs.  .8vo,  2  c."" 
Wright's  Designing  of  Draw-spans : 

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Two  parts  in  one  volume 8vo,  3  50 

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10 

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an  Orifice.     (Trautwine.) 8vo,  2  00 

Bovey's  Treatise  on  Hydraulics 8vo,  5  00 

Church's  Mechanics  of  Engineering 8vo,  6  00 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels payer,  i  50 

Coffin's  Graphical  Solution  of  Hydrauhc  Problems i6mo,  morocco,  2  50 

Plather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  00 

Folwell's  "Water-supply  Engineering 8vo,  4  00 

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8vo,  4  00 

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*  Michie's  Elements  of  Analytical  Mechanics .  .8vo,  4  00 

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Wilson's  Irrigation  Engineering Small  8vo,  4  00 

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Wood's  Turbines 8vo,  2  50 

Elements  of  Analytical  Mechanics , 8vo,  3  00 

MATERIALS  OF  ENGINEERING. 

Baker's  Treatise  on  Masonry  Construction ,  ,  .  .  .Svo,  5  00 

Roads  and  Pavements ,  Svo,  5  00 

Black's  United  States  Public  Works Oblong  4to.  5  00 

Bovey's  Strength  of  Materials  and  Theory  of  Structures Svo,  7  50 

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i6mo,  3  00 

Church's  Mechanics  of  Engineering 8vo,  6  00 

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Johnson's  Materials  of  Construction Large  Svo,  6  oa 

Fowler's  Ordinary  Foundations Svo,  3  50 

Keep's  Cast  Iron Svo,  2  50 

Lanza's  Apphed  Mechanics Svo,  7  50 

Marten's  Handbook  on  Testing  Materials.     (Henning.)     2  vols .8vo,  7  50 

Merrill's  Stones  for  Building  and  Decoration Svo,  5  00 

Merriman's  Text-book  on  the  Mechanics  of  Materials Svo,  4  00 

Strength  of  Materials i2mo,  i  00 

iletcalf' s  Steel.     A  Manual  for  Steel-users i2mo,  3  00 

Patton's  Practical  Treatise  on  Foundations ^ Svo,  s  00 

Richardson's  Modern  Asphalt  Pavements 8vo,  3  oo 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor.,  4  00 

Rockwell's  Roads  and  Pavements  in  France lamo,  i  25 

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Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  00 

Smith's  Materials  of  Machines i2mo,  i  00 

Snow's  Principal  Species  of  Wood 8vo,  3  50 

Spalding's  Hydraulic  Cement i2mo,  2  00 

Text-book  on  Roads  and  Pavements i2mo,  2  00 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  00 

Thurston's  Materials  of  Engineering.     3  Parts 8vo,  8  00 

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Thurston's  Text-book  of  the  Materials  of  Construction 8vo,  5  00 

TiUson's  Street  Pavements  and  Paving  Materials 8vo,  4  00 

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Specifications  for  StL.  i  Bridges i2mo,  i  25 

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the  Preservation  of  Timber 8vo,  2  00 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  00 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel 8vo,  4  00 

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Andrew's  Handbook  for  Street  Railway  Engineers 3x5  inches,  morocco,  i  25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  00 

Brook's  Handbook  of  Street  Raihoad  Location i6mo,  morocco,  i  50 

Butt's  Civil  Engineer's  Field-book i6mo,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  morocco,  i  50 

Railway  and  Other  Earthwork  Tables 8vo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  i6mo,  morocco,  5  00 

Dredge's  History  of  the  Pennsylvania  Railroad:   (1879) Paper,  5  00 

*  Drinker's  Tunnelling,  Explosive  Compounds,  and  Rock  Drills. 4to,  half  mor.,  25  00 

Fisher's  Table  of  Cubic  Yards Cardboard,        25 

Godwin's  Railroad  Engineers'  Field-book  and  Explorers'  Guide.  .  .  i6mo,  mor.,  2  50 

Howard's  Transition  Curve  Field-book i6mo,  morocco,  i  50 

Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Era- 

bankments 8vo,  i  00 

Molitor  and  Beard's  Manual  for  Resident  Engineers i6mo,  i  00 

Wagle's  Field  Manual  for  Railroad  Engineers i6mo,  morocco,  3  00 

Philbrick's  Field  Manual  for  Engineers i6mo,  morocco,  3  00 

Searles's  Field  Engineering i6mo,  morocco,  3  00 

Railroad  SpiraL i6mo,  morocco,  i  50 

Taylor's  Prismoidal  Formula  and  Earthwork 8vo,  i  50 

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Embankments  by  the  Aid  of  Diagrams 8vo,  2  00 

The  Field  Practice  of  Laying  Out  Circular  Curves  for  Railroads. 

^        i2mo,  morocco,  2  50 

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Wellington's  Economic  Theory  of  the  Location  of  Railways Small  8vo,  5  00 

DRAWING. 

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*  Bartlett's  Mechanical  Drawing 8vo,  3  00 

*  "  "  "        Abridged  Ed 8vo,  i  5° 

Coolidge's  Manual  of  Drawing 8vo,  paper  i  00 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  Engi- 
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Diirley's  Kinematics  of  Machines 8vo,  4  00 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo.    2  50 


Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective 8vo,    2  00 

Jamison's  Elements  of  Mechanical  Drawing 8vo,    2  50 

Jones's  Machine  Design : 

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MacCord's  Elements  of  Descriptive  Geometry 8vo,    3  00 

Kinematics ;  or.  Practical  Mechanism 8vo,    5  00 

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Meyer's  Descriptive  Geometry 8vo,     2  00 

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Reid's  Course  in  Mechanical  Drawing 8vo,    2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo, 

Robinson's  Principles  of  Mechanism 8vo, 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo, 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) 8vo, 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing.  i2mo, 

Drafting  Instruments  and  Operations i2<nos 

Manual  of  Elementary  Projection  Drawing i2mo. 

Manual  of  Elementary  Problems  in  the  Linear  Perspective  of  Form  and 

Shadow i2mo, 

Plane  Problems  in  Elementary  Geometry i2mo. 

Primary  Geometry i2mo. 

Elements  of  Descriptive  Geometry,  Shadows,  and  Perspective 8vo, 

General  Problems  of  Shades  and  Shadows 8vo, 

Elements  of  Machine  Construction  and  Drawing 8vo, 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry 8vo, 

Weisbach's  Kinematics  and  Power  of  Transmission.    (Hermann  and  Kllein)8vo, 

Whelpley's  Practical  Instruction  in  the  Ait  of  Letter  Engraving i2mo, 

Wilson's  (H.  M.)  Topographic  Surveying 8vo, 

Wilson's  (V.  T.)  Free-hand  Perspective 8vo, 

Wilson's  (V.  T.)  Free-hand  Lettering , 8vo, 

Woolf's  Elementary  Course  in  Descriptive  Geometry Large  8vo, 


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Voltaic  Cell 8vo, 

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Crehore  and  Squier's  Polarizing  Photo-chronogfraph 8vo, 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  i6mo,  morocco, 
Dolezalek's    Theory   of   the    Lead   Accumulator    (Storage    Battery),      (Von 

Ende.) i2mo, 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo, 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo, 

Gilbert's  De  Magnete.     (Mottelay.) 8vo, 

Hanchett's  Alternating  Currents  Explained i2mo,    i  00 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,    2  50 

Hoknan's  Precision  of  Measurements 8vo,    2  00 

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Kinzbrunner's  Testing  of  Continuous-Current  Machines.  .  , Svo,    2  00 

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Le  Chateliens  High-temperature  Measurements.  (Boudouard — Burgess.)  i2mo,    3  00 
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*  Michie's  Elements  of  Wave  Motion  Relating  to  Sound  and  Light 8vo,  4  00 

Kiaudet's  Elementary  Treatise  on  Electric  Batteries.     (Fishback.) i2mo,  2  50 

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Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  1 8vo,  2  50 

Thurston's  Stationary  Steam-engines 8vo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat 8vo,  i  50 

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Wait's  Engineering  and  Architectural  Jurisprudence 8vo, 

Sheep, 
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Bolland's  Iron  Founder i2mo,  2  50 

"  The  Iron  Founder,"  Supplement i2mo,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  the 

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Eissler's  Modern  High  Explosives 8vo,  4  00 

Effront's  Enzymes  and  their  Apphcations.     (Prescott.) 8vo,  3  00 

Fitzgerald's  Boston  Machinist i2mo,  1  00 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  00 

Hopkin's  Oil-chemists'  Handbook 8vo,  3  00 

Keep's  Cast  Iron 8vo,  2  50 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control Large  8vo,  7  50 

Matthews's  The  Textile  Fibres 8vo,  3  50 

Metcalf' s  SteeL     A  Manual  for  Steel-users i2mo,  2  00 

Metcalfe's  Cost  of  Manufactures — And  the  Administration  of  Workshops. 8vo,  5  00 

Meyer's  Modern  Locomotive  Construction 4to,  10  00 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  1  50 

*  Reisig's  Guide  to  Piece-dyeing. 8vo,  25  00 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  00 

Smith's  Press-working  of  Metals 8vo,  3  00 

Spalding's  Hydraulic  Cement i2mo,  2  00 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses.    ...  i6mo,  morocco,  3  Of> 

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Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  00 

Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
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Ware's  Manufacture  of  Sugar.     (In  press.) 

West's  American  Foundry  Practice i2mo,  2  50 

Moulder's  Text-book i2mo,  2  50 

10 


Wolff's  Windmill  as  a  Prime  Mover 8vo,    3  00 

Wood's  Rustless  Coatings :   Corrosion  and  Electrolysis  of  Iron  and  Steel.  .8vo,    4  00 


MATHEMATICS, 

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*  Bass's  Elements  of  Differential  Calculus i2mo,  4  00 

Briggs's  Elements  of  Plane  Analytic  Geometry 12 mo,  i#oo 

Compton's  Manual  of  Logarithmic  Computations i2mo,  i  50 

Davis's  Introduction  to  the  Logic  of  Algebra 8vo,  i  50 

*  Dickson's  CoUege  Algebra Large  i2mo,  i  50 

*  Introduction  to  the  Theory  of  Algebraic  Equations Large  i2mo,  1  25 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo,  2  50 

Halsted's  Elements  of  Geometry 8vo,  i  75 

Elementary  Synthetic  Geometry. 8vo,  i  50 

Rational  Geometry i2mo,  i  75 

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100  copies  for  5  00 

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10  copies  for  2   00 

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Johnson's  (W.  W.)  Curve  Tracing  in  Cartesian  Co-ordinates i2mo,  1  00 

Johnson's  (W.  W.)  Treatise  on  Ordinary  and  Partial  Differential  Equations. 

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Laplace's  Philosophical  Essay  on  ProbabiUties.     (Truscott  and  Emory.) .  i2mo,  2  00 

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Maurer's  Technical  Mechanics 8v , ,  4  00 

Merriman  and  Woodward's  Higher  Mathematics.  , 8vo,  5  00 

Merriman's  Method  of  Least  Squares 8vo,  2  00 

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Wood's  Elements  of  Co-ordinate  Geometry 8vo,  2  00 

Trigonometry:   Analytical,  Plane,  and  Spherical i2mo,  i  00 


MECHANICAL  ENGINEERING. 

MATERIALS  OF  EKGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Bacon's  Forge  Practice i2mo,  i  50 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Baxtlett's  Mechanical  Drawing 8vo,  3  00 

*  "  "  "        Abridged  Ed 8vo,    1  50 

Benjamin's  Wrinkles  and  Recipes i2mo,    2  00 

Carpenter's  Experimental  Engineering 8vo,     6  00 

Heating  and  Ventilating  Buildings 8vo,    4  00 

Gary's  Smoke  Suppression  in  Plants  using  Bituminous  Coal.     (In  Prepara- 
tion.) 

Clerk's  Gas  and  Oil  Engine Small  8vo,    4  00 

Coolidge's  Manual  of  Drawing 8vo,  paper,     i  00 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
gineers  Oblong  4to,    2  50 

11 


Cromwell's  Treatise  on  Toothed  Gearing lamo,  i  50 

Treatise  on  Belts  and  Pulleys i2mo,  i  50 

Durley's  Kinematics  of  Machines 8vo,  4  00 

Flather's  Dynamometers  and  the  Meastirement  of  Power i2mo,  3  00 

Rope  Driving i2mo,  2  00 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i  25 

Hall's  Car  Lubrication i2mo,  i  00 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Button's  The  Gas  Engine 8vo,  5  00 

Jamison's  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,  1  50 

Part  n.     Form,  Strength,  and  Proportions  of  Parts. 8vo,  3  00 

Kent's  Mechanical  Engineers'  Pocket-book i6mo,  morocco,  5  00 

Kerr's  Power  and  Power  Transmission 8vo,  2  00 

Leonard's  Machine  Shop,  Tools,  and  Methods.     (In  press.) 

Lorenz's  Modern  Refrigerating  Machinery.     (Pope,  Haven,  and  Dean.)     (In  press.) 

MacCord's  Kinematics;   or,  Practical  Mechanism 8vo,  5  00 

Mechanical  Drawing 4to,  4  00 

Velocity  Diagrams 8vo,  i  50 

Mahan's  Industrial  Drawing.     (Thompson.) 8vo,  3  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  00 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  00 

Richard's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  00 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  00 

Smith's  Press-working  of  Metals 8vo,  3  00 

Thurston's   Treatise    on   Friction  and   Lost   Work   in   Machinery  and   Mill 

Work 8vo,  3  00 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics .  i2mo,  i  00 

Warren's  Elements  of  Machine  Construction  and  Drawing .8vo,  7  50 

Weisbach's    Kinematics    and    the    Power    of    Transmission.     (Herrmann — 

Klein.) 8vo,  5  00 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .8vo,  5  00 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  00 

Wood's  Turbines 8vo,  2  50 


MATERIALS   OF    ENGINEERING. 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  ot  the  Materials  of  Engineering.    6th  Edition. 

Reset 8vo,  7  50 

Church's  Mechanics  of  Engineering 8vo,  6  00 

Johnson's  Materials  of  Construction 8vo,  6  00 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.) 8vo,  7  50 

Merriman's  Text-book  on  the  Mechanics  of  Mgterials 8vo,  4  00 

Strength  of  Materials i2mo,  i  00 

Metcalf's  Steel.     A  manual  for  Steel-users i2mo,  '  2  00 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  00 

Smith's  Materials  of  Machines i2mo,  i  00 

Ihm-ston's  Materials  of  Engineering 3  vols.,  8vo,  8  00 

Part  II.     Iron  and  Steel 8vo,  3  So 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents ^vo,  2  5<» 

Text-book  of  the  Materials  of  Construction 8vo,  5  o« 

13 


Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on 

the  PreseiTation  of  Timber 8vo,    2  00 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,    3  00 

food's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

SteeL 8vo,    4  00 


STEAM-ENGINES  AND  BOILERS. 


Berry's  Temperature-entropy  Diagram i2mo,  i  25 

Carnot's  Reflections  on  the  Motive  Power  of  Heat.     (Thurston.) i2mo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book. ..  .i6mo,  mor.,  5  00 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  00 

Goss's  Locomotive  Sparks 8vo,  2  00 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy i2mo,  2  00 

Button's  Mechanical  Engineering  of  Power  Plants 8vo,  5  00 

Heat  and  Heat-engines 8vo,  5  00 

Kent's  Steam  boiler  Economy 8vo,  4  00 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,  i  50 

MacCord's  Slide-valves 8vo,  2  00 

Meyer's  Modern  Locomotive  Construction 4to,  10  00 

Peabody's  Manual  of  the  Steam-engine  Indicator i2mo,  i  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors 8vo,  i  00 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines 8vo,  5  00 

Valve-gears  for  Steam-engines 8vo,  2  50 

Peabody  and  Miller's  Steam-boilers 8vo,  4  00 

Pray's  Twenty  Years  with  the  Indicator Large  8vo,  2  50 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) i2mo,  i  23 

Reagan's  Locomotives:   Simple   Compound,  and  Electric i2mo,  2  50 

Rontgen's  Principles  of  Thermodynamics.     (Du  Bois.) 8vo,  5  00 

Sinclair's  Locomotive  Engine  Running  and  Management i2mo,  2  00 

Smart's  Handbook  of  Engineering  Laboratory  Practice i2mo,  2  50 

%iow's  Steam-boiler  Practice 8vo,  3  00 

Spangler's  Valve-gears 8vo,  2  50 

Notes  on  Thermodynamics i2mo,  i  00 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  00 

Thurston's  Handy  Tables 8vo,  i  50 

Manual  of  the  Steam-engine 2  vols.,  8vo,  10  00 

Part  I.     History,  Structure,  and  Theory 8vo,  6  00 

Part  n.     Design,  Construction,  and  Operation 8vo,  6  00 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake 8vo,  5  00 

Stationary  Steam-engines , .  8vo,  2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice i2mo,  i  50 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation 8vo,  5  00 

Weisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) 8vo,  5  00 

Whitham's  Steam-engine  Design 8vo,  5  00 

Wilson's  Treatise  on  Steam-boilers.     (Flather.) i6mo,  2  50 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines. .  .8vo,  4  00 

MECHANICS  AND  MACHINERY. 

Barr's  Kinematics  of  Machinery gvo,  2  so 

Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Chase's  The  Art  of  Pattern-making i2mo,  2  50 

Church's  Mechanics  of  Engineering 8vo  6  00 

13 


Church's  Notes  and  Examples  in.  Mechanics 8vo,  2  00 

Compton's  First  Lessons  in  Metal-working i2mo,  i  50 

Compton  and  De  Groodt's  The  Speed  Lathe i2mo,  1  50 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  i  50 

Treatise  on  Belts  and  Pulleys i2mo,  i  50 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools.  .  i2mo,  i  50 

Dingey's  Machinery  Pattern  Making i2mo,  2  00 

Dredge's  Record  of  the   Transportation  Exhibits  Building  of  the  World's 

Columbian  Exposition  of  1893 4to  hah  morocco,  5  00 

Du  Bois's  Elementary  Principles  of  Mechanics: 

Vol.      I.     Kinematics 8vo,  3  50 

Vol.    II.     Statics 8vo,  4  00 

Vol.  III.     Kinetics 8vo,  3  50 

Mechanics  of  Engineering.     Vol.    I Small  4to,  7  50 

Vol.  II Small  4t0s  10  00 

Durley's  Kinematics  of  Machines 8vo,  4  00 

Fitzgerald's  Boston  Machinist i6mo,  i  00 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  00 

Rope  Driving i2mo,  2  00 

Goss's  Locomotive  Sparks 8vo,  2  00 

Hall's  Car  Lubrication i2mo,  1  00 

HoUy's  Art  of  Saw  Filing iSmo,  75 

James's  Kinematics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle.  Sm.8vo,2  00 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  00 

Johnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods 8vo,  2  00 

Jones's  Machine  Design: 

Part    I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  00 

Kerr's  Power  and  Power  Transmission 8vo,  2  00 

Lanza's  Applied  Mechanics 8vo,  7  50 

Leonard's  Machine  Shop,  Tools,  and  Methods.     (In  press.) 

Lorenz's  Modern  Refrigerating  Machinery.      (Pope,  Haven,  and  Dean.)      (In  press.) 

MacCord's  Kinematics;  or.  Practical  Mechanism 8vo,  5  00 

Velocity  Diagrams 8vo,  i  50 

Maurer's  Technical  Mechanics 8vo,  4  00 

Merriman's  Text-book  on  the  Mechanics  of  Materials 8vo,  4  00 

*  Elements  of  Mechanics i2mo,  i  00 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  00 

Reagan's  Locomotives:   Simple,  Compound,  and  Electric i2mo,  2  50 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  00 

Richards's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  00 

Ryan,  Norris.  and  Hosie's  Electrical  Machinery.     Vol.  1 8vo,  2  50 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  00 

Sinclair's  Locomotive-engine  Running  and  Management i2mo,  2  00 

Smith's  (0.)  Press-working  of  Metals 8vo,  3  00 

Smith's  (A.  W.)  Materials  of  Machines i2mo,  i  00 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  00 

Thurston's  Treatise  on  Friction  and  Lost  "S/ork  in    Machinery  and    Mill 

Work 8vo,  3  00 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics. 

i2mo,  I  00 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's  Kinematics  and  Power  of  Transmission.   ( Herrmann — -Klein. ) .  8vo ,  5  00 

Machinery  of  Transmission  and  Governors.      (Herrmann — Klein.). 8vo,  5  00 

Wood's  Elements  of  Analytical  Mechanics 8vo,  3  00 

Principles  of  Elementary  Mechanics i2mo,  i  25 

Turbines 8vo.  2  50 

The  World's  Columbian  E^osition  of  1893 4to,  i  00 

14 


METALLURGY. 

t'gleston's  Metallurgy  of  Silver,  Gold,  and  Mercury: 

Vol.    L     Silver 8vo,  750 

Vol.  n.     Gold  and  Mercury 8vo,  7  50 

**  Iles's  Lead-smelting.     (Postage  o  cents  additional.) i2mo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

Le  Chatelier's  High-temperature  Measuremepts.  (Boudouard — Burgess. )i2mo,  3  00 

Metcalf's  Steel.     A  Manual  for  Steel-user& i2mo,  2  00 

Smith's  Materials  of  Machines i2mo,  i  00 

Thurston's  Materials  of  Engineering.     In  Three  Parts 8vo,  8  00 

Part    II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents Svo,  2  50 

Ulke's  Modern  Electrolytic  Copper  Refining Svo,  3  00 

MINERALOGY. 

Barringer's  Description  of  Minerals  of  Commercial  Value,    Oblong,  morocco,  2  50 

Boyd's  Resources  of  Southwest  Virginia Svo  3  00 

Map  of  Southwest  Virignia Pocket-book  form.  2  00 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfipld.) Svo,  4  00 

Chester's  Catalogue  of  Minerals Svo,  paper,  i  00 

Cloth,  I  25 

Dictionary  of  the  Names  of  Minerals Svo,  3  50 

Dana's  System  of  Mineralogy Large  Svo,  half  leather,  12  50 

First  Appendix  to  Dana's  New  "  System  of  Mineralogy." Large  Svo,  i  00 

Text-book  of  Mineralogy Svo,  4  00 

Minerals  and  How  to  Study  Them i2mo,  i  50 

Catalogue  of  American  Localities  of  Minerals Large  Svo,  i  00 

Manual  of  Mineralogy  and  Petrography i2mo.  2  00 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo,  i  00 

Eakle's  Mineral  Tables Svo,  i  25 

Egleston's  Catalogue  of  Minerals  and  Synonyms Svo,  2  50 

Hussak's  The  Determination  of  Rock-forming  Minerals.    (Smith.) -Small  Svo,  2  00 

Merrill's  Non-metaUic  Minerals;   Their  Occurrence  and  Uses Svo,  4  00 

*  P<,nfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

Svo   paper,  o  50 
Roseabusch's   Microscopical   Physiography   ot   the    Rock-making  Minerals. 

(Iddings.") Svo,  5  00 

*  Tillman's  Text-book  of  Important  Minerals  and  Rocks Svo.  2  00 

Williams's  Manual  of  Lithology Svo,  3  00 

MINING. 

beard's  Ventilation  of  Mines l2mo,  2  50 

Boyd's  Resources  of  Southwest  Virginia Svo,  3  06 

Ma.p  of  Southwest  Virginia Pocket  book  form,  2  00 

Douglao's  Untechnical  Addresses  on  Technical  Subjects  .  . i2mo,  i  00 

*Drinier's  Tunneling,  Explosive  Compounds,  and  Rock  Drills.  .4to.  hf.  mor.,  25  00 

Eissler'o  Modern  High  Explosives Svo  4  00 

Fowlei's  Sewage  Works  Analyses 12  mo  2  00 

Goodyear's  Coal-mines  of  the  Western  Coast  of  the  United  States i2mo.  2  50 

Ihlseng's  Manual  of  Mining Svo,  5  00 

**  Iles's  Lead-smelting.     (Postage  gc.  additional.) i2mo.  2  50 

Ktmhardt's  Practice  of  Ore  Dressing  in  Europe Svo.  i  50 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores , Svo,  2  00 

*  Walke's  Lectures  on  Explosives Svo,  4  00 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlcr^nation  Process i2mo,  i  50 

15 


3 

0(7 

4 

OO 

I 

50 

2 

50 

I 

00 

3 

50 

3 

00 

7 

50 

4 

00 

Wilson's  HydrauL.,  and  flacer  Mining i2nio,    2  o» 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation T2mo,     i  25 

SANITARY  SCIENCE. 

Folwell's  Sewerage.     (Designing,  Construction,  and  Maintenance.) 8vo, 

Water-supply  Engineering 8vo, 

Fuertes's  Water  and  Public  Health i2mo. 

Water-filtration  Works i2mo, 

Gerhard's  Guide  to  Sanitary  House-inspection i6nio, 

Goodrich's  Economic  Disposal  of  Town's  Refuse Demy  8vo, 

Hazen's  Filtration  of  Public  Water-supplies 8vo, 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo, 

Mason's  Water-supply.  (Considered  principally  from  a  Sanitary  Standpoint)  Svo , 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2rao,     i  25 

Merriman's  Elements  of  Sanitary  Eng.'neering Svo,    2  00 

Ogden's  Sewer  Design i2mo,    2  00 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo,     i  25 

*  Price's  Handbook  on  Sanitation i2mo,     i  50 

Richards's  Cost  of  Food.     A  Study  in  Dietaries i2mo,     i  00 

Cost  of  Living  as  Modified  by  Sanitaiy  Science i2mo,  i  00 

Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Stand- 
point     Svo,  2  00 

*  Richards  and  Williams's  The  Dietary  Computer Svo,  i  50 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewage Svo,  3  50 

Turneaure  and  Russell's  Public  Water-supplies Svo,  5  00 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  i  00 

Whipple's  Microscopy  of  Drinking-water Svo,  3  50 

WoodhuU's  Notes  on  Military  Hygiene i6mo,  i  50 

MISCELLANEOUS. 

De  Fursac's  Manual  of  Psychiatry.  (Rosanoff  and  Collins.).  ..  .Large  i2mo,  2  50 
Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  Svo,  i  50 

Ferrel's  Popular  Treatise  on  the  Winds Svo.  4  00 

Haines's  American  Railway  Management i2mo,  2  50 

Mott's  Composition,  Digestibility,  and  Nutritive  Value  of  Food.  Mounted  chart,  i  25 

Fallacy  of  the  Present  Theory  of  Sound i6mo,  i  00 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1S24-1S94.  .Small  Svo,  3  00 

Rostoski's  Serum  Diagnosis.     (Bolduan.) i2mo,  i  00 

Rotherham's  Emphasized  New  Testament Large  Svo,  2  00 

Steel's  Treatise  on  the  Diseases  of  the  Dog Svo,  3  50 

Totten's  Important  Question  in  Metrology Svo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  00 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  1  00 

Winslow's  Elements  of  Applied  Microscopy i2mo,  i  50 

Worcester  and  Atkinson.      Small  Hospitals,  Establishment  and  Maintenance; 

Suggestions  for  Hospital  Architecture :  Plans  for  Small  Hospital.  i2mo,  i  25 

HEBREW  AND   CHALDEE  TEXT-BOOKS. 

Green's  Elementary  Hebrew  Grammar i2mo,  i  25 

Hebrew  Chrestomathy 8vo,  2  00 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to    the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco,  5  00 

Lettepis's  Hebrew  Bible 8vo,  2  25 


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